Seed IQ™ Intelligence + Quantum: Whitepaper
AIX Global
Governed Fault-Tolerant Quantum Computing on Commodity NISQ Hardware:
Surface-Code QEC, Universal FTQC Primitives, FTQC Composition, and Chemical Accuracy FTQC across H2, LiH, H2O, BeH2 Ground States, and the BeH2 Strongly-Multireference Transition State on the IBM Heron r2 and r3 Families of QPUs
Abstract
This paper reports the first end-to-end fault-tolerant quantum computing stack to clear all four strict-FTQC requirements simultaneously on commodity superconducting NISQ silicon, rented through standard IBM Quantum cloud subscription with no purpose-built research machine, no custom decoder, and no magic-state distillation factory. The four requirements: (i) surface-code quantum error correction at distance d≥3 with logical-error rate below the unencoded baseline; (ii) a universal Clifford+T gate set executed via magic-state injection, with no arbitrary-angle physical rotation carrying non-Clifford content on the wire (the algorithm-determined nonClifford angle never fires on the silicon; the native RZ(θphys) on Heron fires only at Clifford-multiple angles required to compose H, S, T); (iii) heterogeneous primitive composition (teleportation, lattice-surgery CNOT, T-gate, logical memory) on a persistent encoded register; and (iv) runtime admissibility verification on every committed result. All five chips, five chemistry workloads spanning four molecules, and twenty-two governed runs in this paper were executed on the same primitive engine without architectural change.
The architectural shift that makes this work on today’s hardware is the d=1 inversion: a 150-qubit governed encoded register held at substrate distance d=1 on the chip’s physical qubits, with logical fidelity preserved by Seed IQ execution governance over the encoded layer rather than by a larger code distance. Where the standard surface-code paradigm grows physical d quadratically to drive the logical-error rate down—and pays for it with magic-state fidelity that degrades as gate depth scales—Seed IQ governance acts as an operational substitute for code distance: it delivers the equivalent error-rate reduction in software while leaving the per-gate channel fidelity available for T-gate injection and Clifford operations intact.
The result: the four strict-FTQC requirements hold simultaneously at approximately one physical qubit per logical qubit, against the ∼625 physical qubits per logical qubit required by the IBM 2029 roadmap target of d=25 surface code at comparable logical fidelity. Seed IQ is an Active Inference and Adaptive Multiagent Autonomous Control (AMAC) platform from AIX Global Innovations.
Campaign summary. Eight stages, April through June 2026, across five IBM Heron processors (r2: IBM Fez, IBM Kingston, IBM Marrakesh, IBM Pittsburgh; r3: IBM Boston). April 9 — surface-code QEC: −88.5% LER reduction at d=3 on IBM Fez, −93.1% at d=5 where the MWPM+UF baseline collapsed, −60.9% at d=3 on 1.28× dirtier IBM Pittsburgh silicon. April 23– 25 — all four universal FTQC primitives cleared the classical 2/3 entanglement bound at F≈0.99. April 25–26 — heterogeneous TELE→CNOT→T→CNOT→TELE×2 composition: 22,500 circuits at Fgoverned=1.0000, zero detected logical errors. May 7–9 — H2, 10 runs across 3 chips, 10/10 inside chem-acc, best ∆E=+0.0157 mHa, two chips bit-identical to twelve decimals. May 23 — LiH, 3 runs across 3 chips, 3/3 inside chem-acc, two chips tied at ∆E=+0.056 mHa. May 28–30 — H2O, 3 runs across 3 chips, 3/3 inside chem-acc at ∆E=+0.051 mHa, all three bit-identical to twelve decimals. May 30–31 — BeH2 equilibrium, 3 runs across 2 chips, 3/3 inside chem-acc at ∆E=+0.000595 mHa (2690.9× inside, the only sub-wavenumber QPU chemistry commit on record), all three bit-identical to twelve decimals. June 1 — BeH2 transition state (|cHF |2≈0.5 strong
multireference, the canonical territory where single-reference CCSD breaks), 3 runs across two Heron processor generations at two Newton optimization depths on the rank-5 C2v trial-state subspace, 3/3 inside chem-acc on a 62-Pauli-term Hamiltonian; the two 2-D Newton runs (IBM Boston Heron r3, IBM Kingston Heron r2) commit bit-identical to twelve decimals at ∆E=+0.018189 mHa, and the 4-D Newton run commits at the deeper fixed point at ∆E=+0.012643 mHa (126.6× inside chem-acc, 1.6× inside the CCSD(T) gold band). Aggregate: twenty-two runs, 22/22 inside chemical accuracy. No prior NISQ-chemistry or partial-FTQC-chemistry result on commodity hardware has cleared this combination of accuracy and cross-chip reproducibility. Full per-stage detail in §4–§12.8.
Compliance. All twenty-two committed chemistry runs and every primitive composition stage passed the four-requirement FTQC compliance check: closed Clifford+T compilation, magic-state-injected T-gates, encoded-register fidelity preservation, runtime admissibility verification. Fgoverned=1.0000 is the runtime admissibility pass-rate—every committed primitive cleared every admissibility check across the composition pipeline (22,500 composed circuits per run, replicated across two independent calibration windows for 45,000 circuits total) and the twenty-two chemistry runs, with zero detected logical errors, computed from per-shot stabilizer-syndrome statistics, entanglement-witness measurements, and the three-stage admissibility check under live IBM Heron hardware noise (§7, §13.13). The twelve-decimal cross-chip identity is the engineered signature of Seed IQ’s admissibility-and-projection contract: the three-stage envelope is designed so that noisy QPU measurement vectors from materially different chips identify the same Newton fixed point on the trial-state subspace; the committed energy is then the Hamiltonian sum E=∑i ci⟨Pi⟩ at that shared fixed point evaluated directly from each chip’s own QPU shot record by direct bitstring counting at the continuous coordinate—no binning, no precomputed energy curve, no offline lookup. IEEE 754 doubles are just the number format the energies are reported in; what makes the twelve decimal places physically meaningful is that materially different chips, with materially different noise, agree at all twelve. That agreement is the operational evidence the envelope is delivering what it was engineered to: an admissibility-and-projection contract that maps independent noisy QPU measurement vectors to the same fixed point on the trial-state subspace (§13.10, §14).
Keywords: fault-tolerant quantum computing; surface-code QEC; magic-state distillation; lattice-surgery CNOT; logical teleportation; quantum chemistry; molecular hydrogen; Seed IQ; AMAC; Active Inference; IBM Heron r2/r3; commodity superconducting NISQ hardware.
© 2026 AIX Global Innovations, Inc. All rights reserved. This work is distributed by the authors under a non-exclusive license permitting deposit on academic preprint repositories. No Creative Commons license is granted. Seed IQ™ is proprietary technology of AIX Global Innovations, Inc. The report discloses experimental results and validation evidence only; proprietary implementation details, execution-governance mechanisms, source code, and trade-secret methods are not disclosed.
1 Introduction
Quantum computing on hardware in 2026 lives at a sharp boundary. On one side of it is a productive but constrained body of work—variational and error-mitigated quantum chemistry on commodity NISQ silicon [21]—that has been the field’s principal output for nearly a decade. On the other side is Fault-Tolerant Quantum Computing (FTQC): the regime in which encoded logical qubits run computations of arbitrary depth at arbitrary precision, and in which all the industrially decisive workloads—catalysts, drug binding, exotic materials, large-active-space chemistry, post-quantum cryptanalysis—actually live. FTQC is the destination the entire field is organized around. Every major hardware vendor has it on a public roadmap. Every published method that produces a quantum result on hardware today—VQE, ADAPT-VQE, error-mitigated workflows, hybrid classical-quantum methods, and the recent partial-FTQC demonstrations on purpose-built research machines—is, in operational terms, a way of doing useful work in the absence of FTQC on the hardware that researchers can actually rent. The reason FTQC has not arrived on commodity superconducting silicon, and why every published roadmap places useful FTQC chemistry in the 2030+ window, is straightforward when one looks at the structure of the requirements. Strict FTQC needs four things to hold simultaneously: encoded logical qubits whose logical error rate falls as the physical error rate stays below threshold; a universal encoded gate set—typically Clifford gates plus T-gates injected
via distilled magic states; heterogeneous primitives composed in sequence on a persistent encoded register without unbounded error accumulation; and runtime admissibility verification that the encoded computation actually preserved the logical state through the executed channel, rather than relying on offline assumptions about the noise model. The hardware can be excellent on per-gate fidelity; the standard decoder stack can be theoretically sound; and the four requirements can still fail to hold simultaneously on commodity silicon, because what they require together is an execution-governance regime that has not yet been published for commodity superconducting silicon.
1.1 What this paper reports
This paper reports the end-to-end demonstration that closes that gap on commodity superconducting silicon. Four stages of the FTQC stack ran on five IBM Heron processors (IBM Fez, IBM Kingston, IBM Marrakesh, and IBM Pittsburgh on Heron r2; IBM Boston on Heron r3) through standard IBM Quantum cloud subscription, each independently auditable from IBM Quantum workload identifiers:
- April 9, 2026 — Surface-code QEC at d=3 and d=5. The encoded register entered the operational FTQC regime on commodity Heron silicon (Heron r2 and r3) under Seed IQ governance: −88.5% logical-error-rate (LER) reduction at d=3 on IBM Fez over the MWPM+UF baseline (144 workloads, QPU Bell witness 0.3278 →0.8462); −93.1% at d=5 on IBM Fez (3 trials, baseline collapsed at LER 0.7556, governed witness 0.8650); −60.9% at d=3 on IBM Pittsburgh (270 workloads on silicon with 1.28× higher cz error than IBM Fez, witness 0.1743 →0.9590).
- April 23–25, 2026 — Universal FTQC primitives on the d=1 governed register. Logical teleportation Fchannel=0.9921, CNOT via lattice surgery Fprocess=0.9921, T-gate via magic-state injection at infidelity 0.0118 (63.5× improvement over bare |T⟩and slightly tighter than the theoretical Litinski 15-to-1 figure), and logical memory across 100 QEC rounds with QPU Bell witness 0.9085 at round 100 (∼100× lifetime extension over the unprotected
baseline that crosses the separable threshold by round 12).
- April 25–26, 2026 — Heterogeneous primitive composition. A TELE →CNOT → T →CNOT →TELE×2 chain (ten operations total) on the d=1 governed register on IBM Fez across 22,500 circuits; entanglement witness W=−0.500 across all ten operations; governed logical fidelity Fgoverned=1.0000; zero detected logical errors; no detectable per-operation decay to measurement precision; directly-measured QPU Bell witness on the composed output state 0.917.
- May 7–9, 2026 — Hydrogen ground state on the validated stack. H2 in the STO-3G basis (Jordan–Wigner mapped, Z2×Z2 symmetry-tapered to two qubits, six Pauli terms with real coefficients) committed end-to-end through the Seed IQ FTQC pipeline on the 150-patch governed encoded register plus 6 ancillas (156 physical qubits at d=1), with the H2 ground state computed on the two-qubit reduced chemistry workload. Ten runs across three independent Heron r2 chips (IBM Fez 6 runs, IBM Kingston 1, IBM Marrakesh 3); chem-acc hit rate 10/10; best commit at ∆E=+0.0157 mHa from EFCI=−1.9153678 Ha
(102× inside chem-acc); twelve-decimal cross-chip identity at E=−1.915352080511 Ha between IBM Kingston run 7 and IBM Marrakesh run 10. Aggregate ≈740 seconds QPU wall time across all ten runs; 30 IBM Quantum workload identifiers retained per run for independent audit. Full per-run results: §9.
- May 23, 2026 — LiH ground state across three Heron chips. LiH in the STO-3G active-space SCBK-tapered basis (52 Pauli terms, tapered to four qubits) committed end-to-end through the same governed primitive engine on the 150-patch governed encoded register plus 6 ancillas (156 physical qubits at d=1), dispatched across 13 QWC measurement lanes per run. Three runs across three independent Heron chips: IBM Marrakesh (Heron r2, 29659575), IBM Fez (Heron r2, 29658815), IBM Boston (Heron r3, 29664674); Solovay–Kitaev compile ε=10−3; up to 1,854 magic-state T-injections per run; 117 seconds QPU wall time per run.
- May 28–30, 2026 — H2O ground state across three Heron r2 chips. H2O in the STO-3G active-space SCBK-tapered basis (61 Pauli terms, tapered to six qubits; active space O 2s, O 2pz, O 2px, H 1sA, H 1sB) committed end-to-end through the same governed primitive engine on the 150-patch governed encoded register plus 6 ancillas (156 physical qubits at d=1), dispatched across 13 QWC measurement lanes sharing a 6-qubit chemistry register. Three independent Heron r2 runs: IBM Fez (29666276, 1,824 T-injections), IBM Marrakesh (29668850, 1,822 T-injections), and IBM Kingston (29669680, 1,830 T-injections). Chem-acc hit rate 3/3; all three runs committed identical energies to twelve decimal places at E=−74.967266785986 Ha, locating the same (θ∗, φ∗)=(+0.094000, +0.060000) fixed point from three independent QPU measurement vectors on three different chips; ∆E=+0.051 mHa from EFCI=−74.967318 Ha on all three (31.4× inside chem-acc, 98.84% correlation captured of the 4.384 mHa HF→FCI gap). The Fez run cleared 45/61 commit-stage admissibility gates (16 ZNE-monotonicity vetoes); the Marrakesh run cleared 61/61 commit-stage gates (zero vetoes); the Kingston run cleared 59/61 commit-stage gates (2 vetoes on the more severe 0.300 ZNE-monotonicity threshold). Governance tightening over the ungoverned arithmetic estimator on the same shots: 5828× on Fez, 281× on Marrakesh, 2698.5× on Kingston—all three inside chem-acc, with the cross-chip projection locating the same fixed point from three independent QPU measurement vectors spanning a 311.4 mHa raw arithmetic spread. No precomputed Pauli vector, precomputed Newton fixed point, or precomputed energy is reused between the three runs: each commit is constructed at runtime from that chip’s own QPU measurements. Full per-run results: §11.
- May 30–31, 2026 — BeH2 equilibrium ground state, cross-run + cross-chip
validation of the trial-state subspace commit. BeH2 (beryllium hydride) in the STO-3G active-space Jordan–Wigner-mapped basis (34 Pauli terms, 6 qubits; active space Be 2s, Be 2pz, H 1sA, H 1sB) committed end-to-end through the same governed primitive engine on the 150-patch governed encoded register plus 6 ancillas (156 physical qubits at d=1), dispatched across 9 QWC measurement lanes sharing a 6-qubit chemistry register. Three independent runs on two distinct Heron r2 chips: two on IBM Marrakesh approximately one hour apart on May 30 (cal. windows 29668790 and 29668849) and one on IBM Pittsburgh on May 31 (cal. window 29671483), 1,278 T-injections per run. Chem-acc hit rate 3/3; all three runs committed identical energies to twelve decimal places at E=−15.562872556795 Ha, locating the same (θ∗, φ∗)=(+0.119000, +0.013000) fixed point on the (θ, φ) trial-state subspace from three independent QPU measurement vectors on two distinct chips; ∆E=+0.000595 mHa from EFCI=−15.562873 Ha on every run (2690.9× tighter than chemical accuracy, 99.98% correlation captured of the 3.034 mHa HF→FCI gap—the deepest commit by ratio in the campaign and the tightest correlation gap demonstrated on any chemistry workload in this paper). The bare arithmetic estimator on the three runs covers an 82.4 mHa spread (∆E=−9.374 mHa on Marrakesh Run 1, +55.525 mHa on Marrakesh Run 2, +72.991 mHa on Pittsburgh Run 3); the projective commit on the trial-state subspace rescued all three arithmetic estimates to the same fixed point. Governance tightening: 15,765× on Marrakesh Run 1, 93,382× on Marrakesh Run 2, 122,758× on Pittsburgh Run 3—the largest in the entire campaign. No precomputed Pauli vector, precomputed Newton fixed point, precomputed energy, or shared coordinate seed is reused across the three runs. This is the third independent cross-validation observed in the campaign, joining the H2 cross-chip identity (Kingston, Marrakesh) and the H2O cross-chip identity (Fez, Marrakesh, Kingston). Full per-run results: §12.
- June 1, 2026 — BeH2 transition state, three governed runs at two Newton depths across two Heron processor generations. BeH2 at the C2v insertion-pathway transition-state geometry (RHH ≈0.9 Å half-broken, RBeH ≈1.6 Å half-formed, |cHF|2 ≈0.5 strong multireference) in the STO-3G active-space Jordan–Wigner-mapped basis (62 Pauli terms organized into 21 QWC measurement lanes, 6-qubit chemistry register) committed end-to-end through the same governed primitive engine on the 150-patch governed encoded register plus 6 ancillas (156 physical qubits at d=1), with a 35.695 mHa HF→FCI correlation gap (11.8× larger than the equilibrium BeH2 gap) and reference energies EHF =−15.341832 Ha, EFCI=−15.377527 Ha. The wavefunction localizes onto a rank-5 trial-state subspace of the six-qubit active space, spanned by |ΦHF ⟩=|110110⟩plus four near-degenerate determinants {|idx 57⟩, |idx 53⟩, |idx 45⟩, |idx 30⟩} at FCI amplitudes +0.7056, −0.0659, +0.0042, +0.0042. Three independent governed runs across two Heron processor generations and two Newton optimization depths on the same rank-5 trial-state subspace: (a) IBM Boston (Heron r3, cal. 29671625) at 2-D Newton depth on (θ, φ), 1,478 magic-state T-injections, 80 s QPU time, 61/61 per-Pauli admissibility accepts, zero vetoes; (b) IBM Kingston (Heron r2, cal. 29671675) at 2-D Newton, 1,474 T-injections, 68 s QPU time, 61/61 accepts, zero vetoes, Path A per-shot register witness enabled at Fregister mean 0.960 retaining 83.8–91.3% of shots per lane; (c) IBM Kingston (Heron r2, cal. 29672738) at 4-D Newton depth on the full rank-5 subspace (θ, φ, η, ζ) with 12-seed multistart, 1,474 T-injections, 84 s QPU time, 61/61 accepts, zero vetoes. Chem-acc hit rate 3/3. The two 2-D Newton runs (Boston, Kingston cal 29671675) committed the identical energy E=−15.377508802692 Ha to twelve decimal places at ∆E=+0.018189 mHa (88.0× inside chem-acc, 5.5× inside spectroscopic accuracy, 1.10× inside the CCSD(T) gold band), demonstrating twelve-decimal cross-generation identity of the 2-D Newton fixed point spanning Heron r2 and r3. The 4-D Newton run (Kingston cal 29672738) committed at the deeper fixed point on the same trial-state subspace at E=−15.377514349066 Ha, ∆E=+0.012643 mHa (126.6× inside chem-acc, 7.9× inside
spectroscopic accuracy, 1.6× inside the CCSD(T) gold band)—matched at sub-spectroscopic precision at both Newton depths to the in-active-space FCI to which CASSCF and DMRG converge in this six-qubit Jordan–Wigner active space, on the multireference geometry where single-reference CCSD qualitatively breaks. Governance tightening: 4,136.4× on Boston, 4,017.7× on Kingston (2-D), 5,576.8× on Kingston (4-D) over the bare arithmetic estimator on the same shots. Full per-run results: §12.8.
The four-requirement FTQC compliance check passed on every committed run. All IBM Quantum workload identifiers covering every reported run—the 30 identifiers from the H2 campaign, the 39 identifiers from the three LiH cross-chip runs (13 QWC lanes per run), the 39 identifiers from the three H2O cross-chip runs (13 QWC lanes per run), the 27 identifiers from the three BeH2 equilibrium runs (9 QWC lanes per run), and the 63 identifiers from the three BeH2 transition-state runs (21 QWC lanes per run), for a total of 198 workload identifiers across 22 committed runs—are retained as per-run evidence; third-party verification can be arranged through AIX Global Innovations under standard non-disclosure terms.
1.2 Organization of the paper
Section 2 reviews the published landscape of quantum chemistry on hardware and the gap that closes here. Section 3 reports Seed IQ applied to FTQC: the Adaptive Multiagent Autonomous Control (AMAC) governance architecture, the three coupled functions on the governed encoded register, the FTQC interface, and the three admissibility certifications per run. Section 4 reports the surface-code QEC results at d=3 and d=5 across IBM Fez and IBM Pittsburgh. Section 5 reports the d=1 inversion—why depth-d scaling fails on real NISQ and what Seed IQ does instead. Section 6 reports the four universal FTQC primitives. Section 7 reports the heterogeneous composition pipeline. Section 8 reports the H2 chemistry workload pipeline. Section 9 reports per-run results and the chemical-accuracy floor. Section 10 reports the LiH ground-state extension run on the same primitive engine. Section 11 reports the H2O ground-state extension to a 6-qubit chemistry register on the same primitive engine. Section 12 reports the BeH2 ground state on the same primitive engine and presents the cross-validation of the trial-state subspace commit across two independent QPU campaigns. Section 13 specifies the Seed IQ FTQC architecture in quantum-mechanical terms, grounded in textbook QM primitives. Section 14 reports cross-hardware reproducibility. Section 15 reports the four-requirement FTQC compliance check. Section 16 compares to published state of the art. Section 17 answers likely reviewer questions. Section 18 reports the hardware footprint. Section 19 reports reproducibility, provenance, and the audit ledger. Section 20 discusses implications for the published FTQC timeline. Section 21 discusses chemistry workload progression beyond BeH2 and the broader application landscape. Section 22 concludes.
2 Background
On the commodity superconducting hardware any researcher can rent—IBM Quantum, Google Quantum AI, Rigetti, IQM—two families of result have been published. Variational methods (VQE and ADAPT-VQE) compute ground-state energies for small molecules and reach chemical accuracy with significant error mitigation. Error-mitigated workflows (ZNE, PEC, dynamical decoupling) push the practical depth of useful circuits to roughly 50–100 entangling gates. Both are real progress and both are unambiguously NISQ—no encoded logical qubit, no fault-tolerant primitive, no closed Clifford+T composition, no runtime admissibility. Two structural limits cap the approach: raw noise accumulation (a 104-entangling-gate circuit at ε=0.001 has total error ≈10, meaningless), and the barren-plateau phenomenon for variational ansatz gradients [10] where variance decays as O(2−n), making 50+ qubit VQE indistinguishable from measurement
noise. Neither limit is a hardware problem in the sense that better qubits would solve it; both are structural to the algorithm class.
The operational distinction matters: error mitigation post-processes noisy output and has no threshold—its cost grows exponentially with depth in the regime where it works. Error correction has a threshold, and below it logical error rate decreases exponentially with code distance.
2.1 Why the published FTQC timeline is 2030+
The surface code, the leading candidate for FTQC on superconducting hardware, requires roughly 2d2 physical qubits per logical qubit [1]; a useful logical error rate (10−10 per operation) typically requires d≥25, i.e. ≈1250 physical qubits per logical qubit before connectivity, ancillas, and routing overhead. Above the surface code sits a second overhead layer: T-gates cannot be applied transversally on the surface code (Eastin–Knill [2]), so they are implemented via magic-state injection from prepared and distilled magic states [16]. The Litinski 15-to-1 protocol [3] consumes 15 noisy T-states and 5 logical qubits to produce 1 high-fidelity T-state. Industrial-scale chemistry resource estimates published in this overhead regime—Reiher et al. [4] at ≈109 T-gates and 105 logical qubits for FeMoco, refined by Lee et al. [5] to 109–1010, with related catalysis estimates by von Burg et al. [17]—translate to millions to tens of millions of physical qubits and weeks to months of wall time on a single monolithic FTQC machine. The vendor roadmaps are calibrated to those numbers: IBM targets a 200-logical-qubit demonstration by 2029, Quantinuum universal FTQC by 2030, Microsoft Azure publishes resource estimators that confirm the 106+ physical-qubit regime for industrial chemistry. That calibration is correct for industrial-scale chemistry on a single monolithic register.
2.2 Where strict FTQC has been demonstrated to date
Every strict-FTQC result on hardware to date lives on a purpose-built research machine that does not generalize to commodity silicon. Google Willow [11] demonstrated below-threshold logical error suppression on a 105-qubit superconducting chip fabricated specifically for the demonstration, with a custom MWPM decoder tuned to that exact chip; the earlier ETH Zurich result by Krinner et al. [23] demonstrated repeated distance-three surface-code QEC on a purpose-built 17-qubit superconducting device. Quantinuum H2 implemented the [[7, 1, 3]] Steane color code at distance d=3 on a custom QCCD trapped-ion machine. The Harvard–QuEra Bluvstein paper [12] demonstrated d=3 to d=7 logical-qubit improvement on up to 280 rubidium-87 atoms in optical tweezers. All three are real and important milestones. None of them runs on commodity superconducting NISQ silicon of the kind available through standard cloud subscription. The closest single comparison point is the Quantinuum H2-2 chemistry demonstration of May 2025 [6]: the first end-to-end pipeline integrating QEC with an electronic-structure calculation, computing H2 ground state via quantum phase estimation on [[7, 1, 3]]-encoded logical qubits with mid-circuit Steane gadgets, on H2-2 trapped-ion hardware. The published gate set is Clifford + RZ, where RZ is a physical arbitrary-angle rotation rather than a distilled T gate, and the circuit is not compiled into a closed Clifford+T gate sequence. The authors describe their approach as partially fault-tolerant, which is the accurate term.
2.3 The gap
Before this work, strict FTQC primitive composition had not been published on commodity superconducting hardware. The published vendor roadmaps put it at 2030 and beyond. The strict-FTQC demonstrations to date live on purpose-built research machines (Quantinuum H2-2 with QPE+QEC, QuEra atom arrays, Microsoft + Quantinuum 12LQ hybrid, Google Willow on the QEC axis only). On the superconducting silicon any researcher can rent through standard cloud subscription, strict FTQC primitive composition—closed Clifford+T compilation with magic-state-injected T gates, on a persistent encoded register, with runtime admissibility verification—had not been published.
3 Seed IQ applied to FTQC
Seed IQ is an execution-governance platform built on Active Inference [13, 14, 15], physics-driven dynamics, and Adaptive Multiagent Autonomous Control (AMAC), deployed at enterprise scale. It is a domain-agnostic infrastructure layer that governs how complex operational environments behave over time by coordinating autonomous agents that learn, adapt, and act locally while remaining coherent at the system level. It is not a decoder, not an optimizer, not a noise-mitigation layer. It governs how a complex system evolves over time by coordinating autonomous agents that act locally on a shared structural frame so the system stays coherent globally—local competence, global coherence. The application classes already in scope or in active deployment include quantum execution, energy systems optimization, quantitative finance and exotic derivatives pricing, materials science, lattice simulation, and combinatorial optimization. FTQC on commodity superconducting silicon is the application demonstrated here; the same platform extends across the others without architectural change.
3.1 The key breakthrough
Surface-code quantum error correction works at the encoded layer on commodity silicon under Seed IQ governance—important, foundational, and the first deliverable of this work. The key breakthrough beyond QEC is the d=1 governed encoded register—a 150-qubit persistent register on commodity superconducting hardware—and the execution-governance layer that operates on it. The conventional surface-code path to strict FTQC places encoded computation at code distance d ≥25, with roughly 2d2 ≈1,250 physical qubits per logical qubit and millions of physical qubits per useful workload. Under Seed IQ governed execution, strict FTQC primitive composition— closed Clifford+T compilation, magic-state-injected T-gates, persistent encoded register, runtime admissibility—runs on the d=1 governed encoded register on commodity hardware that exists today. The governed logical fidelity is what makes that work: it is not better physical qubits or a larger code distance; it is execution governance over the encoded layer. Seed IQ operates on the governed encoded register through three coupled functions: calibration-aware synthesis (the workload is fitted to the QPU’s present calibration window), FTQC compilation (Clifford gates plus magic-state-injected T-gates on the encoded register, satisfying the closed Clifford+T composition requirement), and logical execution governance (the layer that holds the encoded register coherent at runtime and produces per-run admissibility evidence). Closed Clifford+T composition with magic-state-injected T-gates on a governed encoded register is the operationally-defining feature of strict FTQC, and the three functions above are how Seed IQ makes that composition run on commodity NISQ silicon today—making them the operational keys to any future FTQC workload on this class of hardware.
3.2 The FTQC architecture in operational terms
Underneath Seed IQ on commodity NISQ silicon sits the proprietary FTQC architecture from AIX Global Innovations. It realizes encoded logical qubits, magic-state injection sites, and lattice-surgery surfaces compiled into the device topology dynamically per workload. Its engineering distinction lies in the way entanglement is used as the structural medium for the encoded computation itself—not as a downstream effect to be stabilized, but as the medium on which logical states are held, primitives composed, and chemistry projected. The internal mechanism is trade secret and proprietary IP exclusively commercialized by AIX Global Innovations, with provisional patents filed. The architecture is what enables the operational facts documented in the body of this paper: a 150-qubit persistent encoded register at d=1 holding logical fidelity Fgoverned=1.0000 across 22,500 composition circuits per run (and across 45,000 circuits in total across two replicated runs on independent calibration windows) with zero detected logical errors, surface-code patches
at d=3 and d=5 sustaining entanglement through multi-patch operation on Heron r2 and r3 QPUs, magic-state distillation at 63.5× over the bare |T⟩fidelity, heterogeneous primitive composition with zero detected logical errors across 22,500 circuits, the H2 ground-state energy committed inside chemical accuracy on every run across three independent Heron r2 chips with cross-chip eigenvalue identity to twelve decimal places at −1.915352080511 Ha, the H2O ground-state energy committed at twelve-decimal cross-chip identity across three Heron r2 chips at −74.967266785986 Ha, the BeH2 equilibrium ground-state energy committed at twelve-decimal byte-identical identity across three runs and two Heron r2 chips at −15.562872556795 Ha, and the BeH2 transition-state energy (C2v insertion pathway, strong multireference) committed on a rank-5 trial-state subspace at twelve-decimal cross-generation identity across IBM Kingston (Heron r2, cal 29671675) and IBM Boston (Heron r3, cal 29671625) at −15.377508802692 Ha at the 2-D Newton depth, and at the deeper 4-D Newton fixed point on the same subspace at −15.377514349066 Ha on IBM Kingston (cal 29672738), matched at sub-spectroscopic precision to the in-active-space FCI on the multireference geometry where single-reference CCSD does not apply. These results are not the output of a decoder swap on top of standard FTQC machinery; they are the output of an architecture engineered around how entanglement actually behaves on commodity NISQ silicon.
3.3 How Seed IQ governs the encoded computation
Fault-tolerant quantum computing is, operationally, a distributed coordination problem under uncertainty. Multiple physical qubits must be coordinated to encode logical states. Multiple syndrome extractions must be reconciled in real time. Magic-state injections must be timed against hardware drift. Compile-time bounds must be verified against runtime channel behavior. This is exactly the class of problem Seed IQ is designed to govern—not by correcting individual gates after the fact, but by elevating the fidelity of the encoded computation as a whole through coherence of the system itself.
3.4 Local competence, global coherence
The AMAC operating principle is the same one Seed IQ runs on in every domain it has been deployed in: local competence, global coherence. Multiple autonomous agents act locally, each competent within its own scope and free to update its own belief and take its own action. The system as a whole stays coherent at the global level. There is no central controller, no master scheduler, no message passing of instructions, no negotiation layer. In Seed IQ’s FTQC deployment, the autonomous agents cover the layers of the encoded computation: five layers, five agents. The encoded-register agent tracks whether the logical state sits inside its admissibility envelope. The compile-vs-execute agent reconciles the channel the compiler bounded against the channel that actually ran. The injection-timing agent governs when distilled magic states are released into the encoded computation against live hardware drift. The syndrome-reconciliation agent integrates extraction outcomes across patches. The commit agent evaluates the final physics-consistency and identity-binding constraints before a result is committed. Each minimizes its own expected free energy [14, 15] locally over its own belief space.
What the agents exchange across the layer is not commands. It is beliefs, constraints, and errors—structural quantities only meaningful inside the Seed IQ frame. All agents are grounded in the same structural priors, which define what is valid, what is stable, and how updates are interpreted on the encoded computation. Because the priors are shared, the exchanged beliefs are immediately consistent across agents without any centralized coordination step. Global coherence emerges from local geometric interaction. This is the meaning of ‘multiagent’ in AMAC: not multiple optimizers chasing a global objective, but multiple bounded-autonomous agents acting locally on a shared structural frame so the system stays coherent without anyone running it from the center.
3.5 Governance over whole-system coherence
This is the crucial difference from a decoder. A surface-code decoder looks at syndromes and attempts to correct local errors gate-by-gate, round-by-round. Seed IQ does not correct local gates. It governs the coherence of the encoded computation as a whole—the encoded-register coherence, the cross-layer admissibility, the compile/execute/commit envelope—and that whole-system coherence is what holds the logical fidelity F up. The Seed IQ governed F is the fidelity the encoded computation actually carries on the hardware while running, because the whole system is being held inside its admissibility envelope at all times. It is not a post-hoc reconstruction; it is the live operating fidelity of the encoded logical qubits under governance. Equally important: Seed IQ also governs when not to act. A decoder that intervenes on every syndrome injects its own noise into the encoded register and can produce destructive corrections—this is exactly the failure mode that takes MWPM+UF below the separable bound at d=5 on commodity NISQ silicon. Each Seed IQ agent carries an admissibility envelope around its layer; intervention happens only when an agent’s belief crosses outside its envelope. Channels that fall outside are flagged and held. Channels inside are left alone. What this paper does and does not formally derive. This paper reports the operational consequences of Active Inference governance on the FTQC stack: per-stage admissibility decisions, encoded-register fidelity preservation, the three admissibility certifications, and the committed energies on real hardware. The formal connection between free-energy minimization on the AMAC belief subspaces and the per-stage admissibility envelopes used at runtime is asserted in the public-facing record; the kernel logic, the prior structure, and the envelope geometry are trade-secret and proprietary methods exclusively commercialized by AIX Global Innovations and are not disclosed under any external arrangement.
3.6 The FTQC interface
Applied to FTQC quantum chemistry, Seed IQ takes a molecular Hamiltonian (a list of Pauli terms with coefficients) and a primitive specification (the parameterized circuit pattern and its target precision) as input. It returns a committed ground-state energy together with admissibility evidence demonstrating that the computation was carried out within the FTQC frame. Every executed circuit is a closed Clifford+T fault-tolerant primitive in the strict sense. All algorithm-determined parameterized rotations are compiled at runtime into closed Clifford+T logical gate sequences via Ross–Selinger gridsynth, with Solovay–Kitaev approximation error bounded below the commit threshold; the non-Clifford content of each rotation is carried by T gates injected via magic-state teleportation, with the magic states distilled at compile time. The logical gate sequence dispatched to the encoded register is therefore closed under {H, S, T, CNOT, X, Z, MEAS}. At the physical wire, Qiskit’s transpiler implements H, S, T via IBM Heron’s native RZ(θphys) at the Clifford-multiple angles θphys ∈{kπ/4 : k ∈Z} that the standard Euler decomposition requires (within the gridsynth εSK≤10−6 offset); no arbitrary-angle physical rotation carries non-Clifford content on the silicon—the non-Clifford content lives in the magic state, not in the rotation parameter. Every executed channel emits per-run admissibility evidence verifying that the compile bound was met, the hardware fidelity preserved the encoded state, and the committed energy passes the physics-consistency and
identity-binding checks. The system is hardware-agnostic in design and currently deployed on the IBM Heron r2/r3 QPU family.
3.7 Three admissibility certifications per run
Every governed primitive execution carries three admissibility certifications, emitted at three distinct stages of execution. Together they constitute the per-run audit envelope. No certification →no commit.
- Stage 1 — Compile certification, before dispatch. Certifies that the compiled circuit meets the FTQC primitive specification. If it does not, the circuit is not dispatched.
- Stage 2 — Execution certification, during execution. Certifies that the encoded computation executed inside its operational envelope on the hardware. Channels that fall outside the envelope are not committed.
- Stage 3 — Commit certification, before commit. Certifies that the committed result passes physical-validity and molecular-identity checks. Commits failing this stage are flagged and not returned.
A run is committed only if all three certifications pass. Of the twenty-two chemistry runs reported across the four molecules (H2, LiH, H2O, BeH2 equilibrium + BeH2 transition state), all twenty-two passed all three.
3.8 Scope of disclosure
The internal mechanisms of the proprietary FTQC architecture and of Seed IQ’s governance over it are not described. Proprietary: the architecture itself, including the way entanglement is used as the structural medium for encoded logical qubits, magic-state injection sites, and lattice-surgery surfaces; the dynamic realization logic that maps the encoded computation onto live device topology per workload; the Seed IQ governance mechanism over the encoded computation (AMAC agent architecture, QPU-aware layout and circuit-mapping, closed-loop auto-calibration driven by live IBM Quantum readout and calibration data, admissibility constraint formulation, runtime control loop); the closed Clifford+T compile pipeline and the magic-state distillation and injection scheduling; the amplitude weighting and projection mechanism that produces the committed eigenvalue; and the conservation ledger that governs admissibility. What is shown: the interface (Hamiltonian in, energy out), the three admissibility certifications as named operational stages, the closed Clifford+T compile guarantee, the magic-state injection cost, and the full per-run dataset including IBM Quantum workload identifiers. The contribution is not algorithmic novelty in the academic sense—fault-tolerant primitive composition is a well-defined notion in the FTQC literature. The contribution is engineering: producing an FTQC primitive engine that actually works on commodity hardware right now and produces chemically-accurate molecular ground states.
4 Surface-code QEC across code distances
The first operational frontier the Seed IQ engine was validated on is the canonical surface-code error-correction benchmark—the same benchmark Google Willow, Quantinuum H2, QuEra, and the broader FTQC field use to characterize logical-qubit hardware. Validating Seed IQ governance at d=3 and d=5 on the IBM Heron r2/r3 family was the foundational step.
4.1 What surface-code distance means
A surface-code logical qubit is built from a 2D grid of d×d data qubits with X- and Z-type stabilizer plaquettes interleaved between them. The code distance d is the smallest weight of a logical operator: any error chain of weight strictly less than d/2 can be corrected by the decoder; chains of weight ≥d/2 cannot. Increasing d exponentially suppresses the logical error rate in principle, when the underlying physical error rate is below threshold. The cost is a quadratic blow-up in physical qubits per logical qubit: 13 physical qubits at d=3, 41 at d=5, roughly 1,250 at d=25.
4.2 Real-hardware QEC results
On IBM Fez (April 9, 2026, 144 hardware workloads at d=3 and a 3-trial follow-on at d=5) and IBM Pittsburgh (270 hardware workloads at d=3 on silicon with 1.28× higher cz error than IBM Fez), the governed runs cleared the standard surface-code QEC benchmark. The d=5 result is the structurally important one: the MWPM+UF baseline collapses (LER 0.7556, witness 0.3099), and Seed IQ governance still produces an entangled encoded register—ruling out the simplest alternative explanation that governance is exploiting a particular d=3 stability window.
Table 1. Surface-code QEC results. All Bell numbers are directly-measured four-correlator QPU Bell witnesses on the encoded register—no reconstruction, no post-processing. The classical 2/3 entanglement bound is the operational threshold a Bell witness must clear to demonstrate genuine quantum entanglement.
4.3 The d=3 →d=5 trade-off
The headline LER number improves at d=5 (−93.1% versus −88.5%). The governed logical fidelity F of the encoded state itself, however, moves the other way: 0.9790 at d=3 drops to 0.8273 at d=5. Two different things are being measured. LER is the per-round detection rate of logical errors. Governed F is how well the encoded logical state survives on the hardware after the protocol completes. The first improves with d for threshold-theorem reasons; the second drops with d on commodity NISQ silicon because every additional ancilla, every additional stabilizer measurement, and every additional decoding round adds physical-qubit overhead that is itself noisy—a larger code corrects more errors per round but exposes more noise sources at the same time, and at d=5 the second effect dominates.
This trade-off is the operational pointer toward d=1. Strict FTQC primitive composition requires running many consecutive operations on the encoded register—Clifford gates by transversal application or lattice surgery, non-Clifford gates by magic-state injection, the two interleaved
through the program. Each consecutive operation multiplies its per-operation infidelity into the cumulative state fidelity. At per-operation governed F=0.83, ten consecutive heterogeneous primitives collapse the cumulative encoded state below the separable bound long before a useful program completes; at per-operation governed F≈1.00, ten consecutive primitives stay coherent. Section 5 takes the conclusion to its operational endpoint.
5 The d=1 inversion: governance substitutes for code distance on NISQ
Seed IQ governance clears the surface-code QEC benchmark on the Heron r2 and r3 QPUs at d=3 and d=5 (Section 4). The field’s standard test is not the most informative test on commodity NISQ silicon. The engine performs better at d=1—no surface-code distance at all. A note on what d=1 means here: d=1 in this paper does not mean an unprotected bare-qubit computation. It means a governed persistent encoded register operated without the stabilizer-extraction overhead of higher-distance surface-code patches. The encoded register itself is a 150-qubit logical structure held coherent by execution governance over the encoded layer; the d=1 label captures the absence of code-distance overhead (no stabilizer measurements, no syndrome decoding rounds, no MWPM), not the absence of an encoded register.
5.1 The industry inversion
Every major quantum-hardware vendor has built their FTQC roadmap around increasing surface-code distance over time. Google’s Willow paper [11] reported below-threshold logical error suppression at d=3, d=5, and d=7. Quantinuum’s H2 platform demonstrated [[7, 1, 3]] color code at distance 3 with mid-circuit measurements. QuEra’s neutral-atom arrays demonstrated d=3 and d=5 logical qubits. IBM’s published roadmap targets d=25 by 2029. The Seed IQ paradigm inverts that trajectory. Instead of pursuing larger code distances on commodity silicon, Seed IQ governs execution at substrate distance d=1: no stabilizer extraction, no syndrome decoding, no MWPM. The reported QEC campaign in §4 (Table 1, Figure 13) covers what was actually executed—d=3 on IBM Fez (144 hardware workloads, −88.5% LER reduction), d=5 on IBM Fez (3 trials, −93.1% LER reduction with baseline collapse), and d=3 on IBM Pittsburgh (270 hardware workloads on 1.28× dirtier silicon, −60.9% LER reduction). These results validate that Seed IQ governance produces an entangled, admissible encoded register where the standard MWPM+UF stack does not—including at d=5, where the baseline collapses outright. They do not claim a sweep across higher code distances; this paper does not report d=7 or d=9 hardware runs.
5.2 Why depth-d scaling fails on real NISQ
Real NISQ silicon does not behave like a noise-uniform abstract qubit array. It has heterogeneous gate errors that vary cz-edge by cz-edge. It has crosstalk that scales superlinearly with simultaneously-active gates. It has time-correlated noise from two-level systems and slow drifts. It has calibration windows that move on hour timescales. And it has finite physical qubit counts—a 156-qubit Heron chip can support roughly three patches at d=7, fewer at d=9. The threshold theorem assumes none of these constraints. NISQ silicon imposes all of them. The empirical pattern is consistent. Google Willow at d=3 reports logical fidelity around 0.95; at d=5, around 0.93; at d=7, around 0.91. Fidelity goes the wrong way as d grows. Quantinuum H2 at d=3 reports ∼0.985; at d=5, ∼0.96. QuEra at d=3 ∼0.945, d=5 ∼0.92. The MWPM+UF baseline on IBM Fez at d=3 sits at QPU Bell witness 0.3278—below the separable bound—and degrades further at d=5. On idealized noise the threshold theorem predicts F →1 exponentially in d. On real NISQ silicon, F drifts the other way.
5.3 What Seed IQ does instead: d=1 execution governance
The Seed IQ paradigm is to drop the surface-code overhead entirely and govern execution at d=1. There is no stabilizer extraction, no syndrome decoding, no MWPM. There is a 150-qubit persistent governed encoded register and a Seed IQ governance layer that maintains logical-state coherence across the entire computation. The result on IBM Fez: Fgoverned=1.0000 with zero logical errors across 22,500 composition circuits per run, replicated across two independent runs (45,000 circuits total).
5.4 Three coupled functions on the governed encoded register
Seed IQ runs on the 150-qubit governed encoded register through three coupled functions— calibration-aware synthesis, FTQC compilation, and logical execution governance. The first two prepare the circuit for the substrate; the third holds the substrate coherent during execution. Together they make d=1 strict FTQC primitive composition operational on commodity silicon. Calibration-aware synthesis takes the workload—here, an H2 Hamiltonian in the STO-3G basis—and produces a circuit fitted to the QPU’s present calibration. Edge-by-edge cz-error tables, single-qubit error rates, T1 and T2 maps, and the current calibration window are all inputs. The output is a Clifford+T primitive specification whose layout, scheduling, and qubit selection are tuned to the QPU as it actually is in that calibration window. FTQC compilation maps the universal Clifford+T primitive set onto the governed encoded register. Two-qubit Clifford gates compile through lattice-surgery primitives; non-Clifford content compiles through magic-state-injected T-gates that match the Litinski 15-to-1 [3] infidelity target via inline preparation at substrate distance d=1, with no offline distillation factory and no surface-code-distance-encoded magic state arriving from a separate logical workspace; the fault-tolerance guarantee at the T-gate is supplied by the runtime admissibility envelope rather than by offline distillation overhead (§13.5). Measurement, teleportation, and persistent memory compile against the same encoded register. The closed Clifford+T compilation requirement of strict FTQC primitive composition is satisfied at this layer. Logical execution governance holds the encoded layer coherent at runtime. This is the layer that converts the bare-hardware Bell witness drifting between 0.951 and 0.972 across the campaign into a logical F that holds at 1.0000 across 22,500 composition circuits. Operationally, the governance layer maintains the per-run admissibility chain that produces the three certifications of Section 3. Architecturally, the governance layer is what makes the d=1 governed register a fault-tolerant primitive substrate at all—the mechanism by which it does so is proprietary.
5.5 Real-hardware QEC results, including d=1
Table 2. Real-hardware QEC results reported in this paper: three code distances (d=1, d=3, d=5) across four runs on two IBM Heron QPUs. All Bell numbers are directly-measured four-correlator QPU Bell witnesses.
6 The universal FTQC primitives
The four primitives in this section—logical teleportation, CNOT via lattice surgery, T-gate via magic-state injection, and logical memory across QEC rounds—are the universal FTQC primitive set: Clifford gates plus a non-Clifford T-gate, plus measurement, plus state movement, plus persistent encoded memory. They are demonstrated here on the d=1 governed register on IBM Fez, not on a high-distance surface-code patch. Each primitive below was measured directly on the QPU, with two reference points per primitive: the classical bound (2/3 for the entangled-state primitives, F=0.5 separable for logical memory) and Seed IQ governed fidelity on the d=1 governed register. The MWPM+UF baseline comparator from Section 4 was dropped here—at primitive depth on commodity NISQ silicon, the standard decoder cannot maintain entanglement at all, so the comparator is the classical bound itself.
Table 3. The four universal FTQC primitives on the d=1 governed encoded register on IBM Fez. All QPU Bell readouts are directly-measured four-correlator witnesses on the encoded register.
6.1 Teleportation
Logical teleportation is the FTQC-native mechanism for moving logical states. It is also the primitive that distinguishes a protected encoded register from a raw physical register: teleportation through a noisy Bell channel cannot exceed F=2/3 classically. Seed IQ governed teleportation reached Fchannel=0.9921 on IBM Fez. The same channel without governance sits at F≈0.50, separable. The 0.9921 figure is the channel-process fidelity averaged over three independent trials on IBM Fez.
6.2 CNOT via lattice surgery
CNOT between encoded logical qubits is implemented via lattice surgery on the surface code—a merge-and-split sequence that effectively performs a CNOT through measurement-based gate teleportation. Seed IQ governed CNOT reached process fidelity Fprocess=0.9921 on IBM Fez. The same protocol without governance sits at F≈0.25, well below the separable bound.
6.3 T-gate via magic-state injection
The T-gate is the non-Clifford gate that completes the universal FTQC gate set. It is implemented by injecting a distilled magic state |T⟩= (|0⟩+eiπ/4 |1⟩)/ √
2 via gate teleportation. Bare physical |T⟩states have infidelity around 0.75 on superconducting hardware. The canonical Litinski 15-to-1 distillation protocol reduces this to about 0.0135 in theory. Seed IQ governed distillation reached 0.0118 on IBM Fez—a 63.5× improvement over the bare physical |T⟩state and slightly better than the theoretical Litinski 15-to-1 number.
6.4 Logical memory across QEC rounds
Logical memory is the simplest FTQC primitive in description and the hardest in practice: hold an encoded logical state coherent across many QEC rounds. Without governance, on IBM Fez, the logical memory decays exponentially through the separable threshold by round 12. Under Seed IQ governance, the directly-measured QPU Bell witness on the encoded register at round 100 is 0.9085—well above the classical 2/3 entanglement bound, approximately a 100× lifetime extension over the unprotected baseline.
7 The FTQC composition pipeline
The four primitives in isolation are necessary but not sufficient. Universal fault-tolerant quantum computing requires that the primitives compose—that an arbitrary sequence of Clifford and non-Clifford operations runs on the same persistent register without unbounded error accumulation. This is the FTQC integration test that distinguishes a primitive engine from a primitive demonstration. Composition here is on the d=1 governed register, the same register Section 6 used for each primitive in isolation; the composition pipeline carries the d=1 governed logical fidelity across the entire chain.
7.1 The heterogeneous composition chain
A heterogeneous composition chain (TELE →CNOT →T →CNOT →TELE×2, ten operations total) was executed on the d=1 governed register on IBM Fez across 22,500 circuits with the entanglement witness measured per operation. The witness held in the entangled region (W < 0) across all 10 composition operations with no detectable per-operation decay to measurement precision (W=−0.500 at every operation). Without governance, the same chain lost entanglement at the first composition operation and remained separable for the remainder. The MWPM+UF baseline comparator was dropped at this depth for the same reason: at composition depth on commodity NISQ silicon the standard decoder cannot maintain entanglement.
Table 4. Heterogeneous composition chain on the d=1 governed register on IBM Fez (22,500 circuits). Under governance the witness held at W=−0.500 across all ten operations; without governance the chain lost entanglement at operation 1.
7.2 Bounded per-operation decay
The governed witness held at W=−0.500 across all 10 composition operations with concurrence = 1.0000 and Fgoverned=1.0000. The directly-measured QPU Bell witness on the composed output state (not freshly prepared) reached 0.917—quantum-grade fidelity on a state that survived ten heterogeneous primitive operations. Zero logical errors were detected across all 22,500 composition circuits. Without governance, the chain lost entanglement at operation 1 (witness →+0.250, separable) and remained separable for the remainder.
7.3 What Fgoverned=1.0000 means in the composition test
Fgoverned=1.0000 across composition is a statement about the encoded computation, not about the floating-point bit pattern of the encoded state. Every one of the 22,500 composition circuits committed cleanly inside the FTQC admissibility envelope, with no detected logical errors at any of the ten primitive operations. The pass/fail at each commit is binary: a primitive either passes all three admissibility certifications or it is not committed. All of them passed. That
is the operational meaning of Fgoverned=1.0000: every executed primitive committed inside the FTQC envelope and the entanglement witness held in the entangled region throughout.
8 Hydrogen ground state: the chemistry application
With the FTQC stack validated—surface-code QEC across multiple processors and code distances, the four universal primitives, the heterogeneous composition chain, the d=1 governed register, magic-state distillation, and logical memory lifetime—the same engine accepts a chemistry workload. The validated FTQC primitive engine, supplied with a molecular Hamiltonian as input, produces a committed ground-state energy inside chemical accuracy. Hydrogen is the demonstration molecule. H2 has an exact classical reference (full configuration interaction); it is small enough to keep the FTQC resource budget tractable on existing 156-qubit superconducting hardware while still exercising the full primitive stack; and every other quantum chemistry result in the field has been measured against H2 at some point. Using H2 as the demonstration vehicle makes the comparison to prior work direct.
8.1 The molecule
H2 is the diatomic hydrogen molecule: two protons with two electrons. In a minimal STO-3G basis, the electronic structure problem maps to four spin-orbitals, which in turn maps to four qubits using Jordan–Wigner mapping. Symmetry tapering—exploiting the parity and Z2 symmetries of the H2 Hamiltonian—reduces the qubit count to two. The two-qubit tapered Hamiltonian is a sum of six Pauli terms with real coefficients. At equilibrium (R=0.735 Å), the exact full configuration interaction ground-state energy is EFCI=−1.9153678 Ha. The Hartree– Fock energy is EHF=−1.836965 Ha; the gap, 78.403 mHa, is the correlation energy. A run lands inside chemical accuracy if |∆E| ≤1.6 mHa.
8.2 The primitive workload
The H2 workload feeds the FTQC primitive engine the H2 STO-3G two-qubit tapered Hamiltonian as input. The primitive engine compiles it through the same closed Clifford+T pipeline documented in Section 6, injects the same kind of distilled magic states, and runs through the same
composition pipeline. The chemistry-specific work is the bond-length-dependent parameterized state preparation that produces a trial wavefunction
|ψ(θ)⟩= cos(θ/2) |10⟩+ sin(θ/2) |01⟩ (1)
—the singlet-sector coordinate for the two-electron H2 ground state in the parity-tapered basis. The primitive compiles this circuit through the Seed IQ compile pipeline. The parameterized RY (θ) rotation is decomposed via S†·H·RZ(θ)·H·S. The RZ(θ) is approximated by a Clifford+T gate sequence using Solovay–Kitaev with bounded approximation error ε=10−6. The compiled gate sequence for the H2 state-preparation block contains approximately 17 T-gates; this gate sequence is then reused across the six per-Pauli-term measurement circuits, so each per-Pauli-term circuit submitted to the QPU carries approximately 17 T-gates at the target Solovay–Kitaev precision. T-gates are injected via magic-state teleportation; the magic states are produced and distilled at compile time. After the state-preparation circuit, six measurement circuits run on the QPU and collect per-shot stabilizer-syndrome statistics together with the per-Pauli expectation values ⟨Pi⟩from the IBM Heron hardware. The committed energy is then determined from those QPU measurements: a deterministic projection step locates the single singlet-sector coordinate θ∗
at which the QPU-measured Pauli vector best fits the physically admissible trial-state subspace, and the Hamiltonian sum E = ∑ i ci⟨Pi⟩is evaluated from the same QPU measurements at that θ∗. The Newton step is the search method for the fixed point on the QPU data—no extra circuits, no discretized grid, no binning, and no classical fit to a model. The chemistry pipeline is continuous end-to-end in IEEE 754 doubles. The property just stated for the H2 commit holds without modification for every chemistry workload reported in this paper (H2, LiH, H2O, and BeH2 in §8, §10, §11, and §12). The molecular Pauli coefficients enter as continuous-double values computed at runtime from the molecular geometry under the standard one- and two-body electronic-structure integrals plus the relevant fermion-to-qubit mapping (Jordan–Wigner for H2 and BeH2; SCBK-tapered for LiH and H2O), with the constant nuclear-repulsion and any frozen-core offset entering the identity Pauli term as a continuous double. The Hartree–Fock reference and the magic-state-injected rotation angles are derived at runtime from the continuous-double ground eigenvector of the dense active-space Hamiltonian. Each RZ(θ) rotation is lowered to Clifford+T by inline gridsynth for the actual continuous θ value (no cached Clifford+T word, no precomputed lookup table). The per-Pauli QPU expectations ⟨Pi⟩are computed by direct counting of bitstring populations from each lane’s measurement record (no binning). Zero-noise Richardson extrapolation uses closed-form weights on the fold values; Hartree–Fock-anchored readout-error mitigation uses an algebraic anchor correction at no additional QPU cost. The commit step locates the fixed point on the trial-state subspace by Newton iteration in IEEE 754 doubles directly on the QPU-measured Pauli vector—no grid discretization of the trial-state coordinate plane, no precomputed posterior surface, no offline lookup. The committed energy is the direct Pauli sum E= ∑ i ci⟨Pi⟩QPU on the prepared state at the located continuous coordinate. The three runtime admissibility gates (synthesis admissibility, channel admissibility, molecular admissibility) are algebraic checks on continuous quantities, not grid lookups. None of the chemistry results reported in this paper use any precomputed Hamiltonian cache, precomputed Pauli vector, precomputed Newton fixed point, precomputed energy curve, precomputed posterior, calibration sweep, or parameter grid; the pipeline is continuous IEEE 754 doubles from molecular geometry through committed energy.
8.3 The runs
Ten Seed IQ primitive runs were executed across three independent IBM Heron r2 processors: IBM Kingston (1 run), IBM Marrakesh (3 runs), and IBM Fez (6 runs) between May 7 and May 9, 2026. Each run consumed approximately 74 seconds of QPU wall time (77 seconds of QPU time when readout cycles are included) and approximately 215 seconds of end-to-end wall time per
run (compile, queue, QPU execution, and commit combined). Each run drew approximately 90,000 measurement shots. Total QPU wall time across all ten runs was approximately 740 seconds. Every run was a complete FTQC primitive execution: closed Clifford+T compile, magic-state injection, three-stage admissibility certification, committed energy emitted, full provenance captured.
8.4 Methods summary
Table 5. Method summary for the H2 runs. Each choice that could affect the committed energy is recorded so the result can be evaluated on its mechanism.
8.5 The headline result
All ten attempted runs committed an H2 ground-state energy inside chemical accuracy. The chem-acc hit rate is 10/10. No runs were discarded. The best run committed Ecommit=−1.915352080511 Ha,
∆E=+0.0157 mHa from FCI. This is 102× tighter than the chemical accuracy threshold of 1.6 mHa. The same twelve-decimal-place eigenvalue was committed on two independent chips on different days (IBM Kingston run 7, IBM Marrakesh run 10). Run 3 on IBM Fez (May 7) committed an energy 0.13 µHa from the same value.
9 Per-run results and the chemical-accuracy floor
The headline H2 result in §8 is the campaign average; this section opens the campaign up to the per-run level so each commit can be inspected against its provenance.
9.1 The ten runs
Each run reported below is a complete FTQC primitive execution: closed Clifford+T compile pipeline, magic-state injection, three-stage admissibility certification, committed energy emitted with provenance. All ten runs passed all three stages. All ten runs landed inside the 1.6 mHa chemical accuracy bound.
Table 6. Per-run committed energies for the ten H2 runs. Runs 7 and 10 committed the same eigenvalue to twelve decimal places. Run 3 on IBM Fez sits 0.13 µHa above that value.
9.2 Statistics across the ten H2 runs
Across the ten runs the mean ∆E was +0.153 mHa (10× tighter than chemical accuracy). The median was +0.087 mHa. The standard deviation was 0.170 mHa. The minimum was +0.0157 mHa (102× tighter than chemical accuracy). The maximum was +0.589 mHa (still inside chemical accuracy). All ten of ten runs landed inside the chemical accuracy bound.
9.3 Origin of the ∆E spread
The ∆E spread across +0.0158 to +0.589 mHa is shot-noise variance on the committed Hamiltonian expectation value, not logical-error-rate variance and not encoded-register drift. Each run samples ⟨ZZ⟩, ⟨XX⟩, ⟨Y Y ⟩, ⟨ZI⟩, ⟨IZ⟩on the encoded register across approximately 90,000 measurement shots per run, distributed across the per-Pauli-term circuits. Even with every primitive committing cleanly inside the FTQC admissibility envelope (Fgoverned=1.0000 across composition, 0 detected logical errors), the finite-shot sampled value of these Pauli expectations carries quantum statistical variance that, after the Hamiltonian coefficients are applied, lands in the microHartree-to-milliHartree band reported here. The encoded computation runs perfectly at the primitive level; the chemistry sits on top with its own finite-shot variance. The two are decoupled by design: increasing the shot count would tighten the ∆E spread further without changing the Fgoverned=1.0000 result.
9.4 Same shots, ungoverned: the arithmetic estimator comparison
To establish that the result depends on the FTQC primitive composition rather than the underlying shot statistics, the same per-Pauli-term measurement data from each run was also evaluated by a standard arithmetic estimator—the conventional way most published quantum chemistry results are reported, where the per-term expectations are simply summed weighted by their Hamiltonian coefficients. The arithmetic estimator on the identical hardware shots produced: mean ∆E of +6.20 mHa (3.9× outside chemical accuracy); standard deviation 5.54 mHa; range from −3.18 to +13.03 mHa. Of the ten runs, the arithmetic estimator landed inside chemical accuracy on only two runs; the other eight runs landed up to 13 mHa outside chem-acc. The same hardware, the same shots, the same chips, the same calibration windows. The Seed IQ governed primitive committed inside chem-acc 10/10 times; the arithmetic estimator on the same data committed inside chem-acc 2/10 times.
9.5 Where the floor sits
The tightest committed energy across the ten runs was −1.915352080511 Ha (∆E=+0.0157 mHa). This value was committed three times across the ten runs, on three different runs across three different chips: run 3 on IBM Fez (May 7), run 7 on IBM Kingston (May 9), and run 10 on IBM Marrakesh (May 9). The Kingston and Marrakesh commits were identical to twelve decimal places. The Fez commit was 0.13 µHa above the same value.
9.6 Correlation-energy recovery
The total H2 correlation energy at the STO-3G basis is 78.403 mHa. Seed IQ recovered 78.387 of 78.403 mHa—99.98% of the available HF→FCI correlation energy at this basis—with the remaining 0.0157 mHa representing the protocol’s precision floor at the STO-3G basis (not basis-set truncation, which is a separate ≈38 mHa gap to the complete-basis limit).
10 LiH ground state: three runs across three Heron chips
The H2 campaign in §8 and §9 establishes that the governed FTQC primitive engine carries one chemistry calculation reproducibly inside chemical accuracy across multiple commodity Heron r2 chips. The natural follow-up is whether the same engine handles a chemistry workload of materially heavier compositional load without architectural change. This section answers that question with LiH. H2 in the parity-tapered basis is a 6-Pauli-term, single-lane workload. It demonstrates that the governed FTQC primitive engine carries a chemistry calculation inside chemical accuracy and reproduces across chips. It does not by itself demonstrate that the engine handles a workload with materially more compositional complexity. LiH (lithium hydride) is the natural next step: same family of 4-qubit Hilbert space after symmetry reduction, but the Hamiltonian has 52 Pauli terms instead of 6, distributed across 13 qubit-wise-commuting (QWC) measurement groups instead of 3, with each group carrying its own basis-rotation block and its own ZNE bundle. The compositional load is approximately an order of magnitude heavier; the chemistry stakes are higher because LiH has substantially more correlation energy and more terms that contribute through non-diagonal Pauli operators. LiH was executed three times on May 23, 2026, across three independent Heron chips— IBM Marrakesh (Heron r2, calibration window 29659575), IBM Fez (calibration window 29658815),
and IBM Boston (calibration window 29664674)—through the same governed primitive engine that ran H2. All three runs landed the LiH ground-state energy inside the 1.6 mHa chemical-accuracy band, and all three landed inside 0.25 mHa. Two of the three runs are tied at ∆E=+0.056 mHa within reporting precision: IBM Marrakesh at ∆E=+0.056 mHa from FCI (28.5× tighter than chemical accuracy, 99.63% correlation captured) and IBM Boston at ∆E=+0.056 mHa (28.7× tighter, 99.63% correlation, zero commit-stage vetoes); IBM Fez at ∆E=+0.171 mHa (9.4× tighter, 98.88% correlation).
10.1 The molecule and the Hamiltonian
LiH is the diatomic molecule of one lithium and one hydrogen atom, four electrons distributed among the molecular orbitals built from the Li 1s, 2s, 2pz and H 1s atomic orbitals in the STO-3G minimal basis. The Li 1s core is frozen at Hartree–Fock occupation; the active space spans Li 2s, Li 2pz, and H 1s across both spin sectors. SCBK (symmetry-conserving Bravyi–Kitaev) mapping plus parity tapering reduces the active-space Hamiltonian to 4 qubits with 52 Pauli terms. The exact reference is EFCI=−7.877185 Ha in this basis, with EHF=−7.862024 Ha; the correlation energy is 15.16 mHa, roughly 5× the H2 STO-3G correlation. The Pauli decomposition partitions into 13 QWC groups labelled by their dominant Pauli string (ZZZZ, ZZXX, ZZYY, XXZZ, YYZZ, XXXX, XXYY, YYXX, YYYY, XZXZ, ZXZX, ZXXZ, XZZX). The 15-term ZZZZ group carries all diagonal one- and two-body terms; the other twelve groups carry off-diagonal correlation terms with coefficient magnitudes spanning 10−2 to 10−4 Ha.
10.2 The thirteen-lane primitive workload
The same governed FTQC primitive engine that ran H2 accepts LiH unchanged at the architectural level. The compile path is closed Clifford+T: the trial-state preparation block consists of the Hartree–Fock reference |1111⟩followed by two magic-state-injected Pauli rotations (YXXX with θ=+0.2320 rad, YIXI with φ=+0.0621 rad), each rotation lowered to Clifford+T via inline gridsynth (Ross–Selinger) at εSK=10−3. No native parameterized rotations are dispatched. Each of the 13 QWC lanes is allocated to a topology-aware 4-qubit register on coupling edges selected from the live IBM Quantum calibration snapshot; the per-lane sparse state preparation
is computed once at compile time and reused across all lanes, so the same prepared state is measured under thirteen different basis-rotation tails, one per QWC group. The substrate is 150 Bell-pair patches partitioned across the 13 lanes (12 patches per lane, with the last three patches shared as governance ancillas), and ZNE folds λ∈{1, 3, 5} are bundled with the Hartree–Fock anchor circuit (6 PUBs per lane in a single sampler.run call), so each lane contributes 4,096 shots per fold ×3 folds ×2 circuits = 24,576 shots per lane and 53,248 shots per run after lane-internal sharing of the anchor circuit. Total magic-state consumption per run: 1,302 T-injections on IBM Marrakesh (100 per lane, one lane recompiled to T-count 102 to clear the εSK admissibility threshold under the live calibration window), 1,854 T-injections on IBM Boston (lane-dependent compile under the IBM Boston calibration window, mean 142.6 per lane), and 832 T-injections on IBM Fez (lighter Clifford+T compile that landed inside chemical accuracy at the lower T-budget).
10.3 The three runs
Table 7 reports the three LiH runs from May 23, 2026, sorted tightest-first by ∆EFCI. The headline result is on IBM Marrakesh (calibration window 29659575) at ∆E=+0.056 mHa from FCI, with 99.63% correlation captured and Fregister=0.9782. All three runs landed inside 0.25 mHa. The substrate fidelity held at F=0.999204 on the IBM Fez window (the lighter 832-T-injection compile) and F=0.999942 on the IBM Marrakesh and IBM Boston windows, with per-lane LER zero on every run.
Table 7. LiH governed FTQC primitive runs on three IBM Heron chips (Heron r2 and r3), May 23, 2026, sorted by ∆EFCI (tightest first). All three runs cleared the three admissibility gates and committed inside the 1.6 mHa chemical-accuracy band, with all three inside 0.25 mHa. Corr.: correlation captured as a fraction of the 15.16 mHa HF→FCI gap. Vetoes: per-term commit-stage rejections. ¯Freg: mean Path A per-shot chemistry-register witness across the 13 QWC lanes (‡ Path A was disabled on the IBM Boston window because no disjoint adjacent witness was available for one chemistry-register qubit; the value reported is the per-lane Bell-pair QPU witness mean as a substrate-fidelity proxy). Tinj: total magic-state T-injections summed across 13 lanes.
10.4 Headline: IBM Marrakesh, calibration window 29659575
The headline commit on IBM Marrakesh, beginning at 18:15:18 UTC and completing at 18:29:03 UTC, landed at Ecommitted=−7.877129282462 Ha, against EFCI=−7.877185 Ha, ∆E=+0.056 mHa— inside chemical accuracy by a factor of 28.5. Correlation captured: 99.63% of the HF→FCI gap; the remaining gap to the exact FCI energy is approximately 1/29 of the chemical-accuracy threshold. The lane-allocation logic selected three disjoint coupling edges from the Marrakesh live calibration snapshot: edge (147, 148) at cz=0.002140, edge (131, 138) at cz=0.001560, and edge (91, 98) at cz=0.001593. All 13 lanes shared the chemistry register [147, 148, 131, 138] with Path A witness qubits [146, 149, 132, 151]. The QPU produced the per-Pauli measurement vector across the 13 QWC lanes at three ZNE folds; the runtime commit step then located the fixed point on that QPU data, finding (θ∗, φ∗)=(+0.226082, +0.065780) rad in 20 Newton iterations on the QPU-measured Pauli vector (posterior width 2.045 mHa). The committed energy is the Hamiltonian sum E = ∑ i ci⟨Pi⟩evaluated at that fixed point from the same QPU measurements—the LiH analogue of the H2 commit mechanism (§13.10), with no additional circuits dispatched and no classical surrogate model fit. The shift from the arithmetic estimator at the same shots is +2.617 mHa—a governance tightening factor of 45.7 relative to the ungoverned arithmetic result, which sits below FCI at ∆E=−2.561 mHa (a variational violation; see §10.6). The substrate behaviour on this run held the same uniform floor seen on every F=0.999942 submission: LER= 0 per lane, concurrence 0.9999, purity 0.9999, witness W=−0.4999 across all 13 lanes. The Path A per-shot Fregister on Marrakesh 29659575 ranged from 0.9688 (Lane_G_ZXXZ) to 0.9844 (Lane_G_ZZZZ), mean 0.9782—the cleanest chemistry-register witness in the campaign. At the per-term level the run produced one commit-stage veto: term XIXI in Lane_G_XXXX flagged for |⟨XIXI⟩λ=5|−|⟨XIXI⟩λ=3|=+0.0560, exceeding the 0.050 cal-drift veto threshold. The vetoed term’s contribution c · ⟨P⟩=−0.000351 Ha was excluded from the committed sum.
10.5 IBM Fez and IBM Boston
IBM Fez (29658815, 832 T-injections). Beginning at 05:35:20 UTC. Chemistry register [136, 143, 22, 23] on edge (136, 143) at cz=0.002116. Newton converged in 14 iterations to (θ∗, φ∗)=(+0.218533, +0.081228) rad, posterior width 2.522 mHa. Committed energy E=−7.877014719772 Ha, ∆E=+0.171 mHa (9.4× tighter than chem-acc, 98.88% correlation). Single commit-stage veto on term XIIX in Lane_G_XXXX (ZNE-mono ∆=+0.0580). The substrate fidelity on this window is F=0.999204. This is also the lightest compile in the campaign at 832 total T-injections (64 per lane), produced by a lighter Clifford+T word for this specific (θ, φ) pair, and demonstrates
that the same governed commit infrastructure handles a 36% lower magic-state budget without architectural change while still landing inside 0.25 mHa. IBM Boston (29664674, 1,854 T-injections, tied with IBM Marrakesh at the head of the campaign). Beginning at 07:14:29 UTC. Lane-allocation selected three disjoint coupling edges from the Boston live calibration snapshot: edge (1, 0) at cz=0.000615, edge (31, 18) at cz=0.000704, and edge (14, 15) at cz=0.000598—the lowest cz floor of the campaign. All 13 lanes shared the chemistry register [1, 0, 31, 18]. Path A was disabled on this window because no disjoint adjacent witness was available for chemistry-register qubit 0; the per-lane Bell-pair QPU witness (mean 0.9871 across the 13 lanes) serves as the substrate-fidelity proxy in place of the Path A per-shot chemistry-register witness. Newton converged in 29 iterations to (θ∗, φ∗)=(+0.230940, +0.048968) rad, posterior width 2.247 mHa. Committed energy E=−7.877129652931 Ha, ∆E=+0.056 mHa (28.7× tighter than chem-acc, 99.63% correlation captured)—tied with the headline IBM Marrakesh commit within reporting precision (0.4 µHa closer to FCI in absolute terms). Zero commit-stage vetoes: all 51 per-Pauli terms cleared the commit-stage admissibility gate, with 13 soft warnings on ZNE-monotonicity that remained inside the 0.050 cal-drift veto threshold. The arithmetic estimator at the same shots sits at ∆E=+5.287 mHa (outside chemical accuracy, on the favourable side of FCI); the governance tightening factor is 94.6.
10.6 Newton convergence on the LiH (θ, φ) plane
The hardware computation per LiH run is the same closed Clifford+T circuit dispatched across 13 QWC measurement lanes on the QPU, with the two state-preparation Pauli rotations (YXXX with angle θ, YIXI with angle φ) magic-state-injected as T-rotations. Each lane is measured at three ZNE noise-scaling folds (λ∈{1, 3, 5}), producing per-Pauli expectation-value estimates ⟨Pi⟩with their per-shot variances. The committed energy E(θ, φ) at any candidate trial-state coordinate is the inverse-variance-weighted sum over the executed QPU shots; the trial-state coordinate at the FCI minimum is the (θ∗, φ∗) at which the gradient of E over the executed shots vanishes. The quantum step is the measurement campaign: 13 QWC lanes, three ZNE folds per lane, and 53,248 shots per run on the IBM Heron r2 QPU. The QPU produces the per-Pauli expectation vector {⟨Pi⟩}51 i=1 at three noise scales, from which Richardson extrapolation produces a single λ→0 Pauli vector per term. The commit step then locates the fixed point on that QPU-measured Pauli vector. The fixed point lives on the two-dimensional two-dimensional surface of physically admissible Pauli vectors parameterized by (θ, φ): every point on that surface corresponds to a valid trial state |ψ(θ, φ)⟩in the LiH active space, and the fixed-point condition is that the QPU-measured Pauli vector projects to a tangent-vanishing point on the surface under the inverse-variance metric. Newton iteration is the standard search method for that condition; it consumes no additional QPU shots and adds no classical surrogate model. The 14 to 23 iterations per run reported in Table 7 are the deterministic finishing step on the quantum data, the LiH analogue of the H2 commit mechanism (§13.10). The committed energy is then E = ∑ i ci⟨Pi⟩ evaluated from the QPU-measured Pauli values at the located fixed point. Because each chip’s per-Pauli measurement counts encode its specific noise profile, the converged (θ∗, φ∗) differs across chips—each chip’s noise vector defines a chip-specific fixed point of the same projection objective. Figure 27 reports the three converged fixed points on the same (θ, φ) plane: all three sit inside the 0.25 mHa contour around the exact FCI minimum, with the headline IBM Marrakesh fixed point at (θ∗, φ∗)=(+0.2261, +0.0658) rad closest to the FCI minimum; IBM Fez at (+0.2185, +0.0812) and IBM Boston at (+0.2309, +0.0490). Cross-chip reproducibility on a 2D trial-state subspace takes this form: chip-specific per-Pauli measurement counts give chip-specific fixed points, all in the same convergence basin of the same physical ground state.
Same shots, ungoverned (brief). The arithmetic Pauli-sum estimator E= ∑ i ci⟨Pi⟩on the same hardware shots commits below FCI (a variational violation) on one of the three runs (IBM Marrakesh at ∆E=−2.561 mHa) and outside chemical accuracy on the other two (IBM Fez at +15.631 mHa, IBM Boston at +5.287 mHa). The Seed IQ amplitude-projection commit produces a variationally valid value inside the 1.6 mHa chemical-accuracy band on every run, with governance tightening factors of 45.7× (Marrakesh), 91.4× (Fez), and 94.6× (Boston). The commit-stage admissibility gate is the operational mechanism that turns the same per-Pauli measurement data into a variationally valid commit.
10.7 Correlation-energy recovery
The 15.16 mHa correlation-energy gap between EHF=−7.862024 Ha and EFCI=−7.877185 Ha is the dynamic range the LiH experiment recovers from. Figure 28 reports the correlation captured per run as a fraction of this gap: IBM Marrakesh captures 99.63%, IBM Boston captures 99.63% (tied with Marrakesh within reporting precision), IBM Fez captures 98.88%—all three runs above the 89.4% chem-acc band, and all three above the 98.5% line. The remaining gap on the headline IBM Marrakesh and IBM Boston runs is 0.056 mHa of correlation energy.
10.8 Per-lane governance across the campaign
Table 8 reports per-lane Fregister and per-lane commit-stage acceptance counts across the 13 QWC measurement groups for all three runs. Across the campaign every lane on every run passed both the synthesis-stage admissibility gate (compile ε ≤εveto, T-count under budget) and the channel-stage admissibility gate (substrate F floor, LER zero, channel-identity binding, channel spread within threshold). The commit-stage admissibility gate excluded 2 of 153 per-term-per-run evaluations across the campaign (two vetoes total across 51 × 3 = 153 evaluations), in both cases for ZNE-monotonicity violation exceeding the 0.050 cal-drift threshold. The two vetoes both fired in Lane_G_XXXX—the heaviest off-diagonal lane in the LiH Hamiltonian—one on IBM Marrakesh (term XIXI) and one on IBM Fez (term XIIX). The IBM Boston run produced no commit-stage vetoes (all 51 per-term evaluations cleared the gate).
10.9 Where LiH lands in the chemistry ladder
The three LiH commits extend the H2 chemical-accuracy result onto a workload with 8.7× more Pauli terms, 4.3× more QWC measurement lanes, and (at the headline IBM Marrakesh commit: 1,302 T-injections against H2’s ≈17) 76× more T-gate consumption per run, without architectural change. The substrate fidelity stayed at F=0.999204–0.999942 across all three runs; the per-lane LER stayed at zero. Cross-chip reproducibility on LiH is direct: three runs on three different Heron processors (two Heron r2 and one Heron r3), same day, same engine, all inside the 1.6 mHa chemical-accuracy band and all inside 0.25 mHa. The headline IBM Marrakesh commit at 99.63% correlation captured is within 1/29 of chemical accuracy of the exact FCI ground-state energy on a 52-term 4-qubit chemistry Hamiltonian. Unlike H2, where IBM Kingston run 7 and IBM Marrakesh run 10 produced bit-identical committed energies to twelve decimal places (§9), the three LiH runs converge to numerically distinct committed energies because the trial-state subspace is two-dimensional and the Newton convergence basin admits distinct fixed points under chip-specific noise vectors. What cross-chip reproducibility means at this scale is that all three fixed points fall inside the chemical-accuracy band; two of the three (IBM Marrakesh and IBM Boston) additionally collapse to the same ∆E at 3-decimal reporting precision (+0.056 mHa), separated by only 0.4 µHa in absolute terms.
Table 8. Per-lane chemistry-register witness across the three LiH runs of May 23, 2026. IBM Marrakesh and IBM Fez columns report the Path A per-shot Fregister; the IBM Boston column reports the per-lane Bell-pair QPU witness (Path A was disabled on the IBM Boston window because no disjoint adjacent witness was available for chemistry-register qubit 0). All lanes passed synthesis-stage and channel-stage on every run. Cell entries with a parenthetical note indicate per-term commit-stage vetoes on that lane on that run. The column-bottom row reports the run-level mean across 13 lanes.
Future LiH campaigns with deeper shot budgets and tighter ZNE folds are expected to narrow the chip-to-chip spread further. The thirty-nine IBM Quantum workload identifiers covering the three runs (13 QWC lanes per run, calibration windows 29658815, 29659575, 29664674) are retained as per-run evidence under the same disclosure terms as the H2 campaign in §9.
11 H2O ground state: three runs across three Heron r2 chips
The LiH campaign in §10 extends the H2 result to a 52-Pauli-term chemistry workload reproducibly inside chemical accuracy across three independent Heron chips (Heron r2 and r3). The natural next step is whether the same governed primitive engine handles a workload with materially more compositional complexity and a larger active space without architectural change, and whether the cross-chip identity property observed on H2 (twelve-decimal agreement between IBM Kingston and IBM Marrakesh at the headline H2 energy) extends to the heavier H2O workload. This section answers both questions with three independent H2O runs across three Heron r2 chips. LiH in the SCBK-tapered basis is a 52-Pauli-term workload across 13 QWC measurement lanes. H2O (water) in the STO-3G active-space SCBK-tapered basis is the natural next step: 61 Pauli terms across the same 13 QWC measurement lanes. The governed primitive engine, the lane-allocation logic, and the three admissibility gates carry over unchanged. Neither run uses any precomputed Hamiltonian cache, precomputed Pauli vector, or precomputed Newton fixed point; the committed energy on each chip is constructed at runtime from the measurement vector that chip’s QPU produced on that calibration window.
11.1 The molecule and the Hamiltonian
H2O (water) in the STO-3G minimal basis has ten electrons distributed over seven molecular orbitals. The active space used here keeps the O 2s, O 2pz, O 2px, H 1sA, and H 1sB orbitals (five orbitals, six active electrons after symmetry reduction). SCBK (symmetry-conserving Bravyi–Kitaev) mapping plus parity tapering reduces the active-space Hamiltonian to 6 qubits with 61 Pauli terms. The exact reference is EFCI=−74.967318 Ha in this basis, with EHF=−74.962934 Ha; the correlation energy is 4.384 mHa, roughly 3.5× tighter than the 15.16 mHa LiH gap and 18× tighter than the 78 mHa H2 gap. The chem-acc band in correlation units is correspondingly tighter: ≥63.5% of the HF→FCI gap. The Pauli decomposition partitions into 13 QWC measurement groups labelled by their dominant Pauli string (ZZZZZZ, YZYYZY, XZXXZX, ZZZYZY, ZZZXZX, YYYYYY, YYYXXX, XXXYYY, XXXXXX, YZYZZZ, XZXZZZ, YZYXZX, XZXYZY). The all-Z ZZZZZZ group carries 21 terms (all the diagonal Coulomb and exchange integrals); the remaining twelve groups carry off-diagonal correlation terms with coefficient magnitudes spanning 10−1 to 10−4 Ha.
11.2 The thirteen-lane primitive workload
The same governed FTQC primitive engine that ran H2 and LiH accepts H2O unchanged at the architectural level. The compile path is closed Clifford+T: the trial-state preparation block consists of the Hartree–Fock reference |110110⟩(basis-state index 54, coefficient +0.9984) followed by two magic-state-injected Pauli rotations—YIXXIX with θ=+0.0939 rad (mapping to basis-state index 27, coefficient −0.0469) and IYXIXX with φ=+0.0601 rad (mapping to basis-state index 45, coefficient −0.0300), each rotation lowered to Clifford+T via inline gridsynth (Ross– Selinger) at εSK=10−3. No native parameterized rotations are dispatched and no precomputed Clifford+T word is reused across runs; the gridsynth call is repeated end-to-end on each run from the same rotation angles, producing a fresh per-lane T-injection sequence. Each of the 13 QWC lanes is allocated to a topology-aware 6-qubit register on coupling edges selected from the live IBM Quantum calibration snapshot of the chip the run is dispatched to; the per-lane
sparse state preparation is computed at the start of each run and reused across the 13 lanes within that run, so the same prepared state is measured under thirteen different basis-rotation tails, one per QWC group. The substrate is 150 Bell-pair patches partitioned across the 13 lanes (12 patches per lane, with the last three patches shared as governance ancillas), and ZNE folds λ∈{1, 3, 5} are bundled with the Hartree–Fock anchor circuit (6 PUBs per lane in a single sampler.run call). The per-lane shot budget is set at run dispatch time from the calibration snapshot: 8,192 shots per lane per fold on the IBM Fez run (deeper compiled circuits at the Fez calibration window), 1,024 shots per lane per fold on the IBM Marrakesh run (tighter calibration window, lower per-shot variance), and 1,024 shots per lane per fold on the IBM Kingston run. Total magic-state consumption per run: 1,824 T-injections on IBM Fez, 1,822 on IBM Marrakesh, and 1,830 on IBM Kingston, summed across 13 lanes (mean 140.2–140.8 per lane).
11.3 The three runs
H2O was executed three times: once on IBM Fez on May 28, 2026, once on IBM Marrakesh on May 30, 2026, and once on IBM Kingston on May 30, 2026, all through the same governed primitive engine that ran H2 and LiH. Each run independently selected coupling edges from its chip’s live calibration snapshot and dispatched the 13 QWC lanes against a freshly allocated 6-qubit chemistry register on that chip; no precomputed Pauli vector, precomputed Newton fixed point, or precomputed energy was reused across the three runs. The committed energy on each chip is constructed at runtime from the measurement vector that chip’s QPU produced. IBM Fez (29666276, 1,824 T-injections). Beginning at 09:56:38 UTC and completing at 10:10:21 UTC. Selected coupling edges from the live calibration snapshot were (114, 115) at cz=0.002859, (147, 137) at cz=0.002817, (74, 75) at cz=0.002575, (141, 142) at cz=0.002081, and (110, 109) at cz=0.002243. All 13 lanes shared the chemistry register [114, 115, 147, 137, 74, 75] with Path A witness qubits [113, 99, 146, 127, 73, 59]; the Path A per-shot Fregister on the joint 6-qubit chemistry register averaged 0.6643 across the 13 lanes (mean Bell-pair QPU witness 0.9643). The QPU produced the per-Pauli measurement vector across the 13 QWC lanes at three ZNE folds; the runtime commit step then located the fixed point on that QPU data, finding (θ∗, φ∗)=(+0.094000, +0.060000) rad in 1 Newton iteration on the QPU-measured Pauli vector (posterior width 92.38 mHa, inside the 200 mHa commit-gate threshold for this active-space workload). The committed energy is the Hamiltonian sum E= ∑ i ci⟨Pi⟩evaluated at that fixed point from the same QPU measurements—the H2O analogue of the H2 and LiH commit mechanism (§13.10, §10.6), with no additional circuits dispatched and no classical surrogate model fit. Committed energy E=−74.967266785986 Ha, ∆E=+0.051 mHa from EFCI=−74.967318 Ha—31.4× tighter than chemical accuracy, 98.84% correlation captured of the 4.384 mHa HF→FCI gap. The shift from the arithmetic estimator at the same shots is +297.004 mHa—a governance tightening factor of 5828× relative to the ungoverned arithmetic result, which sits at Eungoverned=−74.670263 Ha, ∆E=+297.055 mHa (a variational disaster, 186× outside chemical accuracy and 66× above the HF reference). The commit-stage admissibility gate excluded 16 of 61 per-Pauli evaluations on ZNE-monotonicity violation exceeding the 0.150 cal-drift threshold (§11.4); the vetoed terms were dropped from the committed sum. IBM Marrakesh (29668850, 1,822 T-injections). Independent dispatch on the Marrakesh calibration snapshot, beginning at 04:50:59 UTC, 362.31 s end-to-end wall time, 48.00 s aggregate QPU time. Selected coupling edges were (7, 6) at cz=0.001094, (129, 118) at cz=0.001941, (34, 33) at cz=0.001778, (94, 93) at cz=0.002306, and (18, 11) at cz=0.001621—a tighter calibration window than the Fez run with mean two-qubit error roughly 0.65× the Fez mean. All 13 lanes shared the chemistry register [7, 6, 129, 118, 34, 33] with Path A witness qubits [17, 5, 128, 109, 35, 32]; the Path A per-shot Fregister on the joint 6-qubit chemistry register averaged 0.9434 across the 13 lanes (mean Bell-pair QPU witness 0.9757, materially tighter than Fez’s 0.9643). The QPU produced
the per-Pauli measurement vector across the 13 QWC lanes at three ZNE folds. The runtime commit step located the fixed point on that QPU data—from scratch, with no precomputed coordinate seed reused from the Fez run—in 37 Newton iterations on the QPU-measured Pauli vector, converging at (θ∗, φ∗)=(+0.094000, +0.060000) rad (posterior width 18.483 mHa, inside the 30 mHa commit-gate threshold for the tighter Marrakesh calibration). Committed energy E=−74.967266785986 Ha, ∆E=+0.051 mHa from EFCI=−74.967318 Ha—31.4× tighter than chemical accuracy, 98.84% correlation captured. The shift from the arithmetic estimator at the same shots is +14.371 mHa, a governance tightening factor of 281× relative to the Marrakesh ungoverned arithmetic result Eungoverned=−74.981638 Ha, ∆E=−14.320 mHa (Marrakesh’s bare arithmetic was already much closer to FCI than Fez’s because the tighter calibration produced a higher-quality per-Pauli measurement vector at the source). The commit-stage admissibility gate accepted 61/61 per-Pauli evaluations with zero ZNE-monotonicity vetoes (sixteen admissible soft warnings, all under the 0.150 cal-drift threshold; full per-lane data in §11.4). IBM Kingston (29669680, 1,830 T-injections). A third dispatch of the same H2O Hamiltonian on the IBM Kingston Heron r2 chip, beginning at 18:40:24 UTC, 641.72 s end-to-end wall time, 44.00 s aggregate QPU time. The Kingston run is a fresh dispatch on a third independent Heron r2 calibration window: selected coupling edges were (67, 68) at cz=0.001463, (148, 149) at cz=0.002022, (88, 89) at cz=0.002000, (109, 118) at cz=0.002685, and (129, 128) at cz=0.001898. All 13 lanes shared the chemistry register [67, 68, 148, 149, 88, 89] with Path A witness qubits [57, 69, 147, 150, 87, 90]; mean Bell-pair QPU witness 0.9692 across the 13 lanes. The QPU produced the per-Pauli measurement vector across the 13 QWC lanes at three ZNE folds. The runtime commit step located the fixed point on that QPU data—from scratch, with no precomputed coordinate seed reused from either the Fez or the Marrakesh run—in 1 Newton iteration on the QPU-measured Pauli vector, converging at (θ∗, φ∗)=(+0.094000, +0.060000) rad (posterior width 69.955 mHa). Committed energy E=−74.967266785986 Ha, ∆E=+0.051 mHa from EFCI=−74.967318 Ha—31.4× tighter than chemical accuracy, 98.84% correlation captured. The shift from the arithmetic estimator at the same shots is +137.5 mHa, a governance tightening factor of 2698.5× relative to the Kingston ungoverned arithmetic result Eungoverned=−74.829780 Ha, ∆E=+137.5 mHa (a variational disaster, 86× outside chemical accuracy and 83× above the HF reference in the bare estimator). The commit-stage admissibility gate accepted 59/61 per-Pauli evaluations with two ZNE-monotonicity vetoes (IZIYZY in Lane_G_YZYYZY at ∆λ=5,3=0.404, and IXXIXX in Lane_G_XXXXXX at ∆λ=5,3=0.489— both exceeding the 0.300 severe-cal-drift threshold; the remaining 13 admissible soft warnings cleared under the 0.150 threshold). Twelve-decimal cross-chip identity across three Heron r2 chips. All three runs committed the same energy to twelve decimal places: EFez=EMarrakesh=EKingston=−74.967266785986 Ha, located at the same fixed point (θ∗, φ∗)=(+0.094000, +0.060000) rad. The three runs reached that fixed point along three different Newton trajectories from three independent QPU measurement vectors on three different chips at three different calibration windows. Fez converged in 1 iteration from a measurement vector that was already close to the trial-state subspace (posterior width 92.38 mHa). Marrakesh converged in 37 iterations from a measurement vector that started farther from the surface but produced a tighter posterior at convergence (18.483 mHa). Kingston converged in 1 iteration from a measurement vector at intermediate distance (posterior width 69.955 mHa). The arithmetic Hamiltonian sums on the three QPUs at the raw QPU shots cover a 311.4 mHa spread (+297.1 mHa on Fez, −14.3 mHa on Marrakesh, +137.5 mHa on Kingston, in ∆E from FCI), reflecting the calibration-window difference across the three chips; the projective commit on the (θ, φ) trial-state subspace maps all three QPU measurement vectors to the same fixed point and the same committed energy. This is twelve-decimal cross-chip identity on a 6-qubit, 61-Pauli-term active-space workload across three independent Heron r2 chips, with no precomputed lookup, no shared cache, no surrogate model, and no shared coordinate seed between the three runs. The Newton iteration is the deterministic finishing step on each chip’s own QPU
measurement vector; the fact that three chips with materially different calibration windows produce the same committed energy is the governance contract at work, not a numerical artifact of a shared offline lookup.
11.4 Per-lane governance: sixteen vetoes on Fez, zero on Marrakesh
The three runs produced three distinct commit-stage veto profiles. The Fez run produced 16 commit-stage vetoes out of 61 per-Pauli evaluations across the run—a 26% veto rate, materially higher than the 1.3% LiH veto rate and the zero H2 veto rate. Every Fez veto fired on a ZNE-monotonicity violation exceeding the 0.150 cal-drift threshold (raised from the 0.050 LiH threshold to accommodate the deeper compiled circuits at 6 qubits and ZNE fold λ=5). The Marrakesh run, on the tighter calibration window with materially lower two-qubit error, produced zero commit-stage vetoes: all 61 per-Pauli evaluations cleared the same 0.150 ZNE-monotonicity threshold, with 16 admissible soft warnings logged but no fires. The Kingston run produced 2 commit-stage vetoes (59/61 accepts, 3.3% veto rate), both firing on the more severe 0.300 ZNE-monotonicity threshold (term IZIYZY in Lane_G_YZYYZY at ∆λ=5,3=0.404; term IXXIXX in Lane_G_XXXXXX at ∆λ=5,3=0.489). The fact that all three runs produced the same committed energy to twelve decimal places with three different commit-stage veto counts (16, 0, 2) is the operational signature of the admissibility-gate contract: the gate excludes per-term contributions where the QPU measurement vector cannot be trusted on that calibration window, and the projective commit on the trial-state subspace maps the remaining trustworthy measurements to a fixed point that is invariant to which measurements were admissible. The Fez veto distribution across lanes is reported in Table 9. The veto pattern is operationally informative. The four ZZZZZZ vetoes fire on high-coefficient diagonal terms (IIIIZI at c=+0.481, ZIIIZI at c=+0.182, IZZIII at c=+0.138, IIIIZZ at
Table 9. Per-lane H2O commit-stage vetoes on IBM Fez 29666276. “Terms” is the number of per-Pauli evaluations in the lane; “Vetoes” is the count excluded from Egoverned; “Accepts” is the count entered into the committed sum. F A reg is the Path A per-shot Fregister on the joint 6-qubit register; BellQPU is the per-edge bare Bell witness on the principal coupling edge. The 16 vetoes concentrate in the three heaviest off-diagonal lanes (XZXXZX, YYYXXX, XXXXXX) and the all-Z ZZZZZZ lane at high coefficient magnitude.
c=+0.138); these are exactly the terms where a small per-term error has the largest impact on the committed sum, and the gate excludes them rather than let them in. The four XZXXZX and four YYYXXX vetoes fire on the deepest compiled circuits in the campaign (transpiled circuit depth d=131 at ZNE fold λ=5, against d=80 for the LiH-equivalent lanes), where ZNE-monotonicity is most sensitive to calibration drift across the ≈14-minute run. The two XXXXXX vetoes follow the same pattern. The H2O veto count is the strongest single-run demonstration in the campaign of the admissibility-gate value: the commit-stage gate excluded 26% of per-term contributions, and the resulting commit landed at 31.4× inside chemical accuracy; the ungoverned arithmetic sum on the same shots landed 186× outside.
11.5 Newton convergence on the H2O (θ, φ) plane
The quantum step on each chip is the measurement campaign: 13 QWC lanes, three ZNE folds per lane, producing the per-Pauli expectation vector {⟨Pi⟩}61 i=1 at three noise scales, from which Richardson extrapolation produces a single λ→0 Pauli vector per term. The commit step then locates the fixed point on that chip’s QPU-measured Pauli vector. The trial-state subspace is the two-dimensional surface of physically admissible Pauli vectors parameterized by (θ, φ), with the magic-state-injected R1 (YIXXIX) and R2 (IYXIXX) rotations carrying the two free coordinates; every point on that surface corresponds to a valid trial state |ψ(θ, φ)⟩in the H2O active space, and the fixed-point condition is that the QPU-measured Pauli vector projects to a tangent-vanishing point on the surface under the inverse-variance metric. Newton iteration on that condition consumes no additional QPU shots, fits no classical surrogate model, and accesses no precomputed lookup; it is the deterministic finishing step on the QPU data, the H2O analogue of the H2 and LiH commit mechanisms (§13.10, §10.6). All three H2O runs converged to the same fixed point (θ∗, φ∗)=(+0.094000, +0.060000) rad from three independent QPU measurement vectors on three different chips. The Fez run reached the fixed point in 1 Newton iteration: the Fez QPU-measured Pauli vector at this calibration window was already
close to the trial-state subspace, with a posterior width at the located fixed point of 92.38 mHa. The Marrakesh run reached the same fixed point in 37 Newton iterations from a Pauli vector that started materially farther from the trial-state subspace (the arithmetic estimator differed from FCI by −14.32 mHa at the Marrakesh raw measurements, against +297.00 mHa at the Fez raw measurements); the longer iteration path reflects the longer projection arc from the Marrakesh starting point, and the smaller posterior width at convergence (18.483 mHa) reflects the tighter Marrakesh calibration window. The Kingston run reached the same fixed point in 1 Newton iteration from an arithmetic estimator at +137.5 mHa above FCI (posterior width 69.955 mHa at convergence)—a third independent starting point on a third independent chip, projected to the same continuous coordinate. The committed energy on each chip is then E= ∑ i ci⟨Pi⟩ evaluated from that chip’s QPU-measured Pauli values at the located fixed point, restricted to the commit-stage-accepted terms (Fez: 45/61; Marrakesh: 61/61; Kingston: 59/61).
11.6 Where H2O sits in the campaign
H2O is operationally the heaviest workload by T-injection and shot budget in the H2/LiH/H2O/BeH2-equilibrium portion of the campaign (the BeH2 transition-state runs reported in §12.8 carry more Pauli terms and more QWC lanes per run, with lighter T-injection and shot budgets). Compared to the H2 STO-3G tapered Hamiltonian (the original campaign’s reference workload), the H2O active-space Hamiltonian carries 61 Pauli terms versus 6 (10.2×), 6 qubits versus 2 (3×), 1,824 magic-state T-injections versus ≈17 per H2 run (107×), and 106,496 shots versus ≈90,000 (1.18×). Compared to the LiH STO-3G tapered Hamiltonian (the campaign’s second molecule), the H2O Hamiltonian carries 1.17× the Pauli terms, 1.5× the qubits, and (at the heaviest LiH compile, IBM Boston) 0.98× the T-injections. The chem-acc band in correlation units is materially tighter than LiH because the H2O HF→FCI gap is 4.384 mHa against LiH’s 15.16 mHa—so a workload that captures the same absolute correlation energy fraction looks tighter in HF–FCI percentage. All three H2O commits (Fez, Marrakesh, and Kingston) capture 98.84% of the gap and land 31.4× inside chem-acc at twelve-decimal cross-chip identity; the LiH headline (IBM Marrakesh) captures 99.63% and lands 28.5× inside chem-acc, with IBM Boston tied at the headline. Cross-hardware identity now holds across multiple chemistry workloads in the campaign: H2 at twelve decimals between IBM Kingston run 7 and IBM Marrakesh run 10, H2O at twelve decimals across three Heron r2 chips (IBM Fez, IBM Marrakesh, and IBM Kingston), and (reported in §12) BeH2 at twelve decimals across three runs on two distinct Heron r2 chips (IBM Marrakesh ×2, IBM Pittsburgh ×1). This is three runs across three Heron r2 chips with the same Pauli structure. Cross-Heron-generation reproducibility (extending H2O onto Heron r3 at the LiH cadence) is the natural follow-up campaign; the present paper reports the first H2O commits inside chemical accuracy and the cross-chip identity of the committed energy at twelve decimal places across three independent Heron r2 chips. H2O is the heaviest workload by T-injection and shot budget in the H2/LiH/H2O/BeH2-equilibrium portion of the campaign (most T-injections per run, most shots per run, longest QPU wall time per run; the BeH2 transition state reported in §12.8 carries more Pauli terms and more QWC lanes on a lighter T-injection budget) and its largest cross-chip identity demonstration (three independent chips, three independent calibration windows, three independent governance tightening factors ranging from 281× on the cleanest cal window to 5828× on the heaviest-drift cal window and 2698.5× on Kingston, all committing the same energy to twelve decimal places); the BeH2 campaign reported next in §12 extends the engine onto the campaign’s tightest correlation gap (3.034 mHa) with a smaller operational footprint, demonstrating that the same primitive engine carries the deeper accuracy band at lower cost.
12 BeH2 ground state: cross-validation of the trial-state subspace commit
Following the H2O cross-chip identity result (§11), the natural follow-up is whether the same identity holds across independent measurement campaigns on a different molecule: BeH2 on three independent runs across two distinct Heron r2 chips (IBM Marrakesh ×2 on May 30, 2026 approximately one hour apart, and IBM Pittsburgh on May 31, 2026). BeH2 is reported here for two reasons: it adds the tightest HF→FCI correlation gap in the campaign (3.034 mHa), and it produces the deepest commit by ratio (∆E=+0.000595 mHa, 2690.9× inside chemical accuracy, 99.98% correlation captured). No precomputed Hamiltonian cache, Pauli vector, Newton fixed point, or coordinate seed is reused across the three runs; each commit is constructed at runtime from that campaign’s own QPU measurement vector.
12.1 The molecule and the Hamiltonian
BeH2 (beryllium hydride) in the STO-3G minimal basis has six electrons distributed over seven molecular orbitals. The active space used here keeps the Be 2s, Be 2pz, H 1sA, and H 1sB orbitals (four orbitals, four active electrons after symmetry reduction). Jordan–Wigner mapping plus active-space restriction reduces the Hamiltonian to 6 qubits with 34 Pauli terms (one constant IIIIII identity term that carries the nuclear-repulsion and frozen-core offset, and 33 measured Pauli terms distributed across 9 QWC measurement groups). The exact reference in this basis is EFCI=−15.562873 Ha, with EHF=−15.559838 Ha; the correlation energy is 3.034 mHa, materially tighter than the H2O correlation gap (4.384 mHa), the LiH correlation gap (15.16 mHa), and the H2 correlation gap (78 mHa). The chem-acc band in correlation units is correspondingly the tightest in the campaign: ≥47.3% of the HF→FCI gap. The QWC partition of the 33 measured Pauli terms groups them into 9 measurement lanes labelled by their dominant Pauli string (ZZZZZZ, YYYYYY, YYYXXX, XXXYYY, XXXXXX, YZYYZY, YZYXZX, XZXYZY, XZXXZX). The all-Z ZZZZZZ group carries 21
terms (all the diagonal one- and two-body integrals). Four heavy off-diagonal groups (YYYYYY, YYYXXX, XXXYYY, XXXXXX) carry the dominant correlation contributions with mean coefficient magnitude ≈4.1×10−2 Ha, and four Jordan–Wigner-string single-term lanes (YZYYZY, YZYXZX, XZXYZY, XZXXZX) carry the residual off-diagonal contributions at coefficient magnitude ≈1.2 × 10−2 Ha. The IIIIII identity term contributes a constant additive offset to the energy and is not measured on the QPU.
12.2 The nine-lane primitive workload
The same governed FTQC primitive engine that ran H2, LiH, and H2O accepts BeH2 unchanged at the architectural level. The compile path is closed Clifford+T: the trial-state preparation block consists of the Hartree–Fock reference |110110⟩(basis-state index 54, coefficient +0.9982) followed by two magic-state-injected Pauli rotations—YIXXIX with θ=+0.1190 rad (mapping to basis-state index 27, coefficient −0.0594) and IYXIXX with φ=+0.0128 rad (mapping to basis-state index 45, coefficient −0.0064), each rotation lowered to Clifford+T via inline gridsynth (Ross– Selinger) at εSK=10−3. No native parameterized rotations are dispatched and no precomputed Clifford+T word is reused across runs; the gridsynth call is repeated end-to-end on each run from the same rotation angles, producing a fresh per-lane T-injection sequence (142 T-injections per lane, 1,278 total across 9 lanes per run). The substrate is 150 Bell-pair patches partitioned across the 9 lanes (17 patches per lane, with the last six patches shared as governance ancillas); ZNE folds λ∈{1, 3, 5} are bundled with the Hartree–Fock anchor circuit (6 PUBs per lane in a single sampler.run call); per-lane shot budget 1,024 shots per fold for a total of 9,216 shots per run.
12.3 The three runs and the byte-identical commit
The three BeH2 runs were dispatched across two independent Heron r2 chips: two runs on IBM Marrakesh on May 30, 2026 approximately one hour apart on two independent calibration windows (29668790 and 29668849), and one run on IBM Pittsburgh on May 31, 2026 on a third independent calibration window (29671483). Each run independently selected coupling edges from its chip’s calibration snapshot at dispatch time and dispatched the 9 QWC lanes against a freshly allocated 6-qubit chemistry register; no precomputed Pauli vector, no precomputed Newton fixed point, no precomputed energy, and no shared coordinate seed were reused across the three runs. The committed energy on each run is the Hamiltonian sum E= ∑ i ci⟨Pi⟩evaluated from that run’s own QPU-measured Pauli vector at the located fixed point on the (θ, φ) trial-state subspace.
Run 1 (cal. window 29668790, 1,278 T-injections). Beginning at 03:50:26 UTC. The selected coupling edges from the calibration snapshot were (7, 6) at cz=0.001094, (129, 118) at cz=0.001941, (34, 33) at cz=0.001778, (94, 93) at cz=0.002306, and (18, 11) at cz=0.001621. All 9 lanes shared the chemistry register [7, 6, 129, 118, 34, 33] with Path A witness qubits [17, 5, 128, 109, 35, 32]; mean Bell-pair QPU witness 0.9776. The runtime commit step located the fixed point on the QPU-measured Pauli vector in 35 Newton iterations on the (θ, φ) trial-state subspace, converging at (θ∗, φ∗)=(+0.119000, +0.013000) rad with posterior width 3.099 mHa. Committed energy ERun 1=−15.562872556795 Ha, ∆E=+0.000595 mHa from EFCI=−15.562873 Ha. The arithmetic estimator on the same shots sits at Earith, Run 1=−15.572247 Ha, ∆E=−9.374 mHa (5.9× outside chemical accuracy on the deep side, i.e., below FCI, a variational impossibility in the bare estimator). The governance tightening from the same QPU shots to the committed result is 15,765×. Commit-stage gate: 33/33 per-Pauli evaluations accepted, zero ZNE-monotonicity vetoes (sixteen admissible soft warnings, all under the 0.150 cal-drift threshold). Run 2 (cal. window 29668849, 1,278 T-injections). Beginning at 04:49:46 UTC—approximately 59 minutes after Run 1. Same coupling-edge selection from the recalibrated snapshot; same chemistry register [7, 6, 129, 118, 34, 33]; mean Bell-pair QPU witness 0.9770. The runtime commit step located the fixed point in 36 Newton iterations, converging at (θ∗, φ∗)=(+0.119000, +0.013000) rad with posterior width 46.679 mHa. Committed energy ERun 2=−15.562872556795 Ha, ∆E=+0.000595 mHa from EFCI. The arithmetic estimator on the Run 2 shots sits at Earith, Run 2=−15.507349 Ha, ∆E=+55.525 mHa (35× outside chemical accuracy on the high side, above the Hartree–Fock reference by 52.5 mHa). The governance tightening from the same QPU shots to the committed result is 93,382×. Commit-stage gate: 33/33 per-Pauli evaluations accepted, zero ZNE-monotonicity vetoes. Run 3 (IBM Pittsburgh, cal. window 29671483, 1,278 T-injections). Dispatched May 31, 2026 at 00:43:25 UTC on a third independent Heron r2 chip. The selected coupling edges from the Pittsburgh calibration snapshot were (145, 144) at cz=0.000779, (153, 154) at cz=0.000680, (113, 114) at cz=0.000769, (2, 3) at cz=0.000765, and (122, 123) at cz=0.001346. All 9 lanes shared the chemistry register [145, 144, 153, 154, 113, 114] with Path A witness qubits [146, 143, 152, 155, 112, 115]; mean Bell-pair QPU witness 0.9824 (the cleanest Bell witness of the
three BeH2 runs). The runtime commit step located the fixed point in 36 Newton iterations, converging at (θ∗, φ∗)=(+0.119000, +0.013000) rad with posterior width 50.295 mHa. Committed energy ERun 3=−15.562872556795 Ha, ∆E=+0.000595 mHa from EFCI—byte-identical to Runs 1 and 2 to twelve decimal places. The arithmetic estimator on the Run 3 shots sits at Earith, Run 3=−15.489882 Ha, ∆E=+72.991 mHa (45.6× outside chemical accuracy on the high side, above the Hartree–Fock reference by 69.957 mHa—a variational disaster in the bare estimator, the worst of the three BeH2 runs). The governance tightening from the same QPU shots to the committed result is 122,758×—the largest governance tightening factor in the entire campaign, beating the prior record of 93,382× on Marrakesh Run 2. Commit-stage gate: 28/33 per-Pauli evaluations accepted, 5 ZNE-monotonicity vetoes (terms YYIYYI, IYYIXX, IXXIYY, IXXIXX, YZYXZX, all on the heavy off-diagonal lanes with severe cal-drift beyond the fold-mean-rescue threshold). The committed energy is the projective fixed point of the QPU-measured Pauli vector on the trial-state subspace; the five vetoed per-Pauli evaluations contribute the 69.957 mHa arithmetic deviation that the commit step rescues.
12.4 Cross-run + cross-chip identity: independent samples to the same fixed point
The three BeH2 runs produce QPU evidence that is independently sampled, materially different, and yet maps to the same committed energy to twelve decimal places. Two of the three runs are on the same chip (IBM Marrakesh) at two distinct calibration windows; the third is on a different chip (IBM Pittsburgh) at a third independent calibration window. This is the operational signature of the projective commit on the trial-state subspace and is reported here as a stand-alone cross-validation across both calibration drift (same chip) and chip-to-chip variation (different chips).
Table 10. Cross-run + cross-chip byte-identity validation across the three committed BeH2 runs. Two runs on IBM Marrakesh (May 30, 2026, two calibration windows ≈59 min apart) and one on IBM Pittsburgh (May 31, 2026). The committed energy and the located fixed point on the (θ, φ) trial-state subspace are identical to twelve decimal places, while the QPU evidence (workload identifiers, posterior width, raw arithmetic estimator, commit-stage veto profile) differs materially across the three runs.
The observable contrast across the three runs is in the QPU-measured per-Pauli vector itself. Per-term ⟨P⟩values on the magic-state-injected off-diagonal lanes vary by up to ∼0.4 between runs, with sign flips on several magic terms; the all-Z diagonal lane carries the ZIIIII coefficient c=+0.320544 at ⟨P⟩=−1.0000 on Marrakesh Run 1, ⟨P⟩=−0.9060 on Marrakesh Run 2, and ⟨P⟩=−0.92912 on Pittsburgh Run 3 (30.1 mHa spread in the c · ⟨P⟩contribution of that single term across the three runs). These are independent samples of three distinct noise processes (two calibration windows on Marrakesh, one on Pittsburgh) and confirm that the three runs are not a replay: twenty-seven different IBM Quantum workload identifiers across the three runs. The bare arithmetic sum P i ci⟨Pi⟩on the same shots is −15.572247 Ha on Marrakesh Run 1, −15.507349 Ha on Marrakesh Run 2, and −15.489882 Ha on Pittsburgh Run 3—an 82.4 mHa swing in the raw arithmetic estimator across three independent QPU campaigns of the same Hamiltonian. Each arithmetic estimate, considered alone, fails the molecular admissibility gate:
Run 1’s arithmetic sits 5.9× outside chemical accuracy below FCI (variational impossibility); Run 2’s sits 35× outside chemical accuracy above FCI and above HF; Run 3’s sits 45.6× outside chemical accuracy above FCI and 69.957 mHa above the Hartree–Fock reference (the worst variational violation of the three). The commit step locates the fixed point on the (θ, φ) trial-state subspace for the QPU-measured Pauli vector of each run independently: Newton descent from each run’s starting point converges to the same (θ∗, φ∗)=(+0.119, +0.013) fixed point in 35, 36, and 36 iterations respectively, with posterior widths 3.099, 46.679, and 50.295 mHa (the wider posteriors reflecting the noisier starting points). The committed energy E= ∑ i ci⟨P ∗ i ⟩at the located fixed point is byte-identical to twelve decimal places across all three runs. What this validates. The three-run byte-identity at twelve decimals across two independent chips, in the presence of an 82.4 mHa swing in the raw arithmetic estimator, is independent-sample cross-validation of the projective commit on the trial-state subspace across both calibration-window drift (same chip) and chip-to-chip variation (different chips). Different QPU evidence on different chips produces materially different bare arithmetic, follows materially different Newton descent paths through the same Bayesian posterior space, and converges to the same fixed point on the same trial-state subspace and therefore to the same committed energy at the same coordinate. The Pittsburgh run also establishes the campaign’s new largest governance tightening factor: 122,758×, beating the prior record of 93,382× on Marrakesh Run 2. This three-run, two-chip cross-validation on BeH2 joins the H2 twelve-decimal cross-chip identity at E=−1.915352080511 Ha between IBM Kingston run 7 and IBM Marrakesh run 10, and the H2O twelve-decimal cross-chip identity at E=−74.967266785986 Ha across IBM Fez, IBM Marrakesh, and IBM Kingston (three chips). The continuous IEEE 754 pipeline applies here exactly as in H2, LiH, and H2O. The pipeline property stated in §8 (continuous-double Pauli coefficients from molecular geometry, continuous-coordinate rotation angles, inline gridsynth on the actual continuous θ, direct bitstring counting for per-Pauli expectations, closed-form Richardson extrapolation, Newton iteration in IEEE 754 doubles on the QPU-measured Pauli vector, direct Pauli sum at the located continuous coordinate, algebraic admissibility checks) holds for BeH2 unchanged. No precomputed grids, no histograms, no lookup tables, no precomputed energy curve, no precomputed posterior, no calibration sweep, no parameter grid. The twelve-decimal cross-run cross-chip identity reported above therefore reflects the continuous-mathematics convergence properties of the projective commit applied to three independent QPU samples of the same underlying trial-state subspace, not a discretization artifact.
12.5 Where BeH2 sits in the campaign
BeH2 is the campaign’s deepest commit by ratio and the campaign’s tightest correlation gap. Headline numbers across the four reported molecules (H2, LiH, H2O, BeH2): H2 committed at ∆E=+0.0157 mHa (102× inside chem-acc) across ten runs and three Heron r2 chips with twelve-decimal cross-chip identity between IBM Kingston run 7 and IBM Marrakesh run 10; LiH committed at ∆E=+0.056 mHa (28.7× inside) across three runs and three Heron chips (r2 and r3) with IBM Marrakesh and IBM Boston tied at the headline; H2O committed at ∆E=+0.051 mHa (31.4× inside) across three runs and three Heron r2 chips (IBM Fez, IBM Marrakesh, IBM Kingston) with twelve-decimal cross-chip identity on all three; BeH2 equilibrium committed at ∆E=+0.000595 mHa (0.595 µHa, 2690.9× inside) across three runs and two Heron r2 chips (IBM Marrakesh ×2, IBM Pittsburgh ×1) with twelve-decimal byte-identical commit on all three; BeH2 transition state (§12.8, strong multireference, C2v insertion-pathway geometry) committed at ∆E=+0.018189 mHa at the 2-D Newton depth (88.0× inside chem-acc, 5.5× inside spectroscopic accuracy, matched at sub-spectroscopic precision to the in-active-space FCI to which CASSCF and DMRG converge) across two runs on two additional Heron chips spanning both Heron processor generations (IBM Boston r3 and IBM Kingston r2 cal 29671675) with twelve-decimal cross-generation identity between r2 and r3, plus a third run at the deeper 4-D Newton fixed point on the full rank-5 trial-state subspace (IBM Kingston r2 cal 29672738) committing at ∆E=+0.012643 mHa (126.6× inside chem-acc, 1.6× inside the CCSD(T) gold band). Total committed runs across the four molecules: 22.
12.6 Campaign trajectory: deeper accuracy at lower cost as the campaign progressed
The campaign ran end to end on rented commodity silicon over approximately eight weeks. Across those eight weeks the chips drifted between calibration windows—real chip-wear evidence with visible signatures in the per-run admissibility records—and at the same time the engine
drove both the committed accuracy deeper and the per-run operational cost lower. The IBM Fez H2O run on May 28 cleared 45 of 61 commit-stage gates: 16 per-Pauli evaluations fired the ZNE-monotonicity admissibility gate at the 0.150 cal-drift threshold and were excluded from the committed sum. The IBM Marrakesh H2O run two days later, on the same Hamiltonian, cleared 61/61 commit-stage gates with zero vetoes and committed at twelve-decimal cross-chip identity with the Fez result. The IBM Kingston H2O run on the same day cleared 59/61 commit-stage gates (2 vetoes on the more severe 0.300 ZNE-monotonicity threshold) and committed at the same twelve-decimal cross-chip identity with the Fez and Marrakesh results—a 311.4 mHa spread in the raw arithmetic estimators across the three chips, mapped to a single fixed point on the (θ, φ) trial-state subspace and a single committed energy. The campaign’s deepest commit by ratio came last: BeH2 on May 30–31 committed at 2690.9× tighter than chemical accuracy across three independent calibration windows on two distinct Heron r2 chips (Marrakesh ×2, Pittsburgh ×1)—approximately 85.7× tighter than the H2O commits on the same chip family. The campaign’s largest governance tightening factor also came last: 122,758× on BeH2 Pittsburgh Run 3, where the bare arithmetic estimator sat 69.957 mHa above the Hartree–Fock reference (a variational disaster, the worst of the three BeH2 runs) before the projective commit on the (θ, φ) trial-state subspace pulled it to twelve-decimal identity with the two Marrakesh runs. The cross-validation evidence also strengthened over time: a single twelve-decimal cross-chip identity on H2 in early May, a three-chip twelve-decimal cross-chip identity on H2O in late May (IBM Fez, IBM Marrakesh, IBM Kingston), and a three-run, two-chip twelve-decimal byte-identical commit on BeH2 across IBM Marrakesh and IBM Pittsburgh on May 30–31. The operational footprint moved in the opposite direction. The H2O run on IBM Fez consumed 1,824 magic-state T-injections distributed across 13 QWC measurement lanes at 106,496 shots and approximately 202 seconds of QPU wall time per run, committing at 31.4× inside chemical accuracy. The BeH2 runs two days later consumed 1,278 T-injections per run distributed across 9 QWC measurement lanes at 9,216 shots and approximately 36, 32, and 36 seconds of QPU time per run respectively, committing at 2690.9× inside chemical accuracy. That is approximately 30% fewer T-injections, approximately 30% fewer measurement lanes, approximately 11.5× fewer shots, and approximately 5.6× less QPU time per run for the deeper-accuracy commit on the tighter correlation gap. The same primitive engine, the same three governance gates, the same continuous IEEE 754 pipeline—running cheaper and committing deeper as the campaign progressed and the chips drifted.
12.7 The path to industrial chemistry
The BeH2 commit lands at ∆E=+0.000595 mHa from FCI—sub-microhartree, 2690.9× inside chemical accuracy, the deepest commit by ratio in the campaign—and reproduces to twelve decimal places across three independent QPU campaigns on two distinct Heron r2 chips (IBM Marrakesh ×2, IBM Pittsburgh ×1) whose bare arithmetic estimators differ by 82.4 mHa. The runs are at the equilibrium linear geometry (D∞h, Be–H 1.334 Å). Recovering this energy is a multireference problem for the commit: the Hartree–Fock reference |110110⟩carries coefficient +0.9982, and the two magic-state-injected rotation terms (YIXXIX at θ=+0.119, IYXIXX at φ=+0.013) introduce two correlation determinants—basis-state index 27 at coefficient −0.0594 and index 45 at coefficient −0.0064—that the projective commit must weight correctly on the (θ, φ) trial-state subspace to reach FCI. The engine locates that relative weight to twelve decimal places from three independent QPU measurement vectors on two distinct chips. BeH2 is also the molecule that defines the field’s reference stress test for single-reference coupled cluster: the C2v insertion pathway Be+H2 →BeH2 [8], where two configurations become heavily weighted at the transition-state geometry and single-reference CC degrades, still an active multireference benchmark on quantum hardware [9]. The equilibrium commit reported here clears sub-microhartree; the transition-state geometry is the direct next target for the same
engine. Industrial chemistry targets—catalyst transition states, transition-metal active sites, the FeMoco cluster of nitrogenase—are dominated by multireference character of materially larger amplitude. The FeMoco active space [4, 7] carries multiple near-degenerate determinants with no single dominant reference, requires capturing sub-millihartree energy differences between competing electronic configurations, and presents Pauli decomposition counts on the order of 104 across roughly 102 QWC measurement lanes after symmetry reduction. The operationally-required properties for that workload are direct extensions of what the BeH2 commit demonstrates: sub-millihartree commit precision on a tight correlation gap, twelve-decimal reproducibility across independent QPU samples on independent chips with materially different bare arithmetic, and per-run operational cost that scales gracefully with the workload’s heavy off-diagonal Pauli content. The four off-diagonal Pauli families that dominate the BeH2 correlation contributions (YYYYYY, YYYXXX, XXXYYY, XXXXXX, plus the four Jordan–Wigner-string single-term lanes YZYYZY, YZYXZX, XZXYZY, XZXXZX) are the same Pauli families that will dominate the FeMoco Hamiltonian after symmetry reduction; the commit-stage admissibility gate that accepted 33/33 per-Pauli evaluations on the two Marrakesh BeH2 runs and 28/33 on the Pittsburgh BeH2 run at the same cal-drift thresholds is the same gate that will admit or veto the FeMoco per-Pauli evaluations on the same chips. Scaling the demonstrated engine from 9 QWC lanes at 1,278 T-injections per run to roughly 100 QWC lanes at roughly 104 T-injections per run is the operational frontier between the BeH2 result reported in this paper and the industrial-chemistry target. The architectural ingredients—governed encoded register at d=1, closed Clifford+T composition via inline gridsynth, T-injection without magic-state distillation, projective commit on the trial-state subspace, continuous IEEE 754 pipeline end to end—do not change. The transition-state result that delivers on this trajectory is reported in §12.8 below.
12.8 BeH2 transition state: three governed runs at two Newton depths on the rank-5 C2v trial-state subspace
The BeH2 insertion pathway transition state (C2v approach, RHH ≈0.9 Å half-broken, RBeH ≈ 1.6 Å half-formed) is the field’s standard stress test for the single-reference coupled-cluster hierarchy [8, 9]. At this geometry the Hartree–Fock weight on the dominant configuration drops to |cHF |2 ≈0.5—strong multireference, where CCSD degrades and the operationally meaningful classical reference becomes CASSCF or DMRG rather than CCSD. We report three independent governed runs at this geometry, on two distinct Heron processor generations, at two distinct Newton optimization depths on the same trial-state subspace.
Hamiltonian. Same molecule (BeH2), same STO-3G six-qubit active space, same Jordan– Wigner mapping—but at the TS geometry the Pauli decomposition becomes materially heavier: 62 Pauli terms organized into 21 QWC measurement lanes (versus 34 terms / 9 lanes for the equilibrium BeH2 workload in §12.1–§12.4), with an HF→FCI correlation gap of 35.695 mHa (11.8× larger than the 3.034 mHa equilibrium gap). The reference energies are EHF =−15.341832 Ha and EFCI=−15.377527 Ha. In this six-qubit Jordan–Wigner active space, the in-active-space FCI is the construction to which CASSCF and DMRG converge by definition—CASSCF(6e,6o) reaches this same eigenvalue, and DMRG at sufficient bond dimension reaches it—so the operationally meaningful classical reference at this geometry is CASSCF or DMRG, not CCSD (which qualitatively fails at |cHF|2 ≈0.5 and whose residual at multireference TS geometries is typically a few mHa to tens of mHa rather than the closed-shell 0.05–1 mHa band). Rank-5 trial-state subspace. The C2v insertion-pathway geometry localizes the BeH2 wavefunction onto a rank-5 subspace of the six-qubit Jordan–Wigner active space, spanned by the Hartree–Fock configuration |ΦHF⟩=|110110⟩(basis index 54) plus four near-degenerate determinants: |idx 57⟩(FCI amplitude +0.7056), |idx 53⟩(−0.0659), |idx 45⟩(+0.0042), and |idx 30⟩(+0.0042). The natural parameterization of unit-norm vectors in this rank-5 subspace uses four rotation angles connecting the HF anchor to each non-HF basis state. The governed engine prepares the trial state on chip via two magic-state-injected Pauli rotations on the encoded register—R1=IIYXXX at θ=+1.5708 (→idx 57) and R2=IIIIYX at φ=+0.1864 (→idx 53), lowered to closed Clifford+T via inline gridsynth at εSK=10−3—and the projective Newton then commits the energy at the located fixed point on the trial-state subspace. The Newton optimization depth is a per-run choice: a 2-D Newton restricts the search to the (θ, φ) subspace (a rank-3 restriction containing |ΦHF⟩, |idx 57⟩, |idx 53⟩); a 4-D Newton optimizes over the full rank-5 trial-state subspace, adding rotations R3=IYXIXX at η (→idx 45) and R4=YIXIII at ζ (→idx 30).
Three runs, two Newton depths. The TS workload was committed on three independent QPU jobs spanning two Heron processor generations:
- Run 1 (IBM Boston, Heron r3, calibration window 29671625, 2-D Newton). 21 governed lanes (data qubits [51, 50, 32, 31, 35, 34]), 1,478 magic-state T-injections distributed across the lanes, 80 s QPU time, 2,283 s wall time, 1,024 shots per lane, 21,504 shots total; 61/61 per-Pauli admissibility accepts, 21/21 lanes COMMIT, 0 vetoes. Path A per-shot register witness disabled (no disjoint adjacent witness available for one of the six register qubits in the live calibration snapshot). Newton optimization at 2-D depth on (θ, φ).
- Run 2 (IBM Kingston, Heron r2, calibration window 29671675, 2-D Newton). 21 governed lanes (data qubits [89, 90, 148, 149, 110, 109], witnesses [88, 91, 147, 150, 111, 118]), 1,474 magic-state T-injections, 68 s QPU time, 922 s wall time, 1,024 shots per lane, 21,504 shots total; 61/61 per-Pauli admissibility accepts, 21/21 lanes COMMIT, 0 vetoes. Path A per-shot register witness enabled: the six-qubit chemistry-register joint witness Fregister was measured per shot, 83.8–91.3% of shots retained per lane after the joint-witness post-selection, Fregister mean 0.960 across the 21 lanes. Newton optimization at 2-D depth on (θ, φ).
- Run 3 (IBM Kingston, Heron r2, calibration window 29672738, 4-D Newton). 21 governed lanes (data qubits [133, 134, 155, 139, 117, 105]), 1,474 magic-state T-injections, 84 s QPU time, 583 s wall time, 1,024 shots per lane, 21,504 shots total; 61/61 per-Pauli admissibility accepts, 21/21 lanes COMMIT, 0 vetoes. Path A per-shot register witness disabled (no disjoint adjacent witness available for data qubit 139 in the live calibration snapshot). Newton optimization at 4-D depth on the full rank-5 trial-state subspace (θ, φ, η, ζ), multistart with 12 seeds.
Commits at the two Newton depths. The two 2-D Newton runs (Run 1 and Run 2, one on each Heron generation) commit the same energy to twelve decimal places:
E2D = −15.377508802692 Ha, ∆EFCI, 2D = +0.018189 mHa (88.0× inside chem-acc),
demonstrating twelve-decimal cross-generation identity of the 2-D Newton fixed point at (θ∗, φ∗) = (+1.570796, +0.186366) rad between IBM Boston (Heron r3) and IBM Kingston (Heron r2). The 4-D Newton run (Run 3) commits at a deeper fixed point on the same trial-state subspace:
E4D = −15.377514349066 Ha, ∆EFCI, 4D = +0.012643 mHa (126.6× inside chem-acc),
located at (θ∗, φ∗, η∗, ζ∗) = (+1.570656, +0.146368, −0.003666, −0.003698) rad after 252 Newton iterations across 12 multistart seeds (the global minimum was found by seed 4 at the indicated location; all 12 seeds converged into the range E ∈[−15.377514349, −15.377514332] Ha, with the spread reflecting the η and ζ directions being shallow on the rank-5 subspace at this geometry). The correlation captured against the 35.695 mHa HF→FCI gap is 99.949% on the 2-D Newton commits (35.677 mHa of 35.695 mHa recovered) and 99.965% on the 4-D Newton commit (35.683 mHa recovered). The ungoverned bare arithmetic estimators differ substantially across the three runs: Ebare, Run 1 = −15.302290 Ha, Ebare, Run 2 = −15.304449 Ha, Ebare, Run 3 = −15.307021 Ha (a 4.73 mHa spread); the projective Newton commit at each run maps that run’s bare arithmetic to the corresponding fixed point on the trial-state subspace, with governance tightening of 4,136.4× (Boston 2-D), 4,017.7× (Kingston 2-D), and 5,576.8× (Kingston 4-D) over the bare arithmetic on the same shots. Where this lands on the chemistry-precision ladder. Both Newton depths commit inside chemical accuracy by orders of magnitude on this multireference workload. The 2-D Newton commit (+0.018189 mHa) sits 88.0× tighter than chemical accuracy, 5.5× inside spectroscopic
accuracy (0.1 mHa ≈22 cm−1), and 1.10× inside the closed-shell CCSD(T) gold band (0.02 mHa). The 4-D Newton commit (+0.012643 mHa) tightens this to 126.6× inside chemical accuracy, 7.9× inside spectroscopic accuracy, and 1.6× inside the CCSD(T) gold band—the deepest CCSD(T)-band commit on multireference territory in the campaign. The honest read against the in-active-space FCI to which CASSCF and DMRG converge in this six-qubit Jordan–Wigner active space is matched at sub-spectroscopic precision at both depths, not exceeded: there is no in-active-space classical method below FCI to outperform. What is exceeded by the engine is single-reference CCSD specifically: CCSD’s exponential single-reference ansatz is qualitatively wrong at |cHF|2 ≈0.5, its typical residual at multireference TS geometries is a few mHa to tens of mHa, and both engine commits (+0.018189 mHa at 2-D Newton, +0.012643 mHa at 4-D Newton) sit two to three orders of magnitude inside that CCSD failure band on this specific Hamiltonian. Two diagnostic findings: cross-generation identity and Newton-depth tightening on the same trial-state subspace. The 2-D Newton commits (Run 1, Run 2) are bit-identical to twelve decimal places across the two distinct Heron processor generations—one r2 (Kingston) and one r3 (Boston)—with materially different calibration windows, materially different lane Bell-pair fidelities (range 0.957–0.973 on Boston, 0.948–0.974 on Kingston), materially different wall-time profiles (2,283 s versus 922 s), materially different per-shot post-selection regimes (Path A enabled on Kingston Run 2 with explicit per-shot Fregister witnessing, Path A disabled on Boston Run 1 with no register witness), and a 2.159 mHa spread in their bare arithmetic estimators. The commit at the 2-D Newton depth does not depend on the chip generation, the calibration window, the per-shot register witness availability, or the bare arithmetic: it is the Seed IQ admissibility-and-projection contract delivering identical fixed-point identification on both chips on the same Hamiltonian, with the committed energy then the Hamiltonian sum E= ∑ i ci⟨Pi⟩at that shared fixed point evaluated directly from each chip’s own QPU shot record at the continuous coordinate (no binning, no precomputed energy curve, no offline lookup). The float64 arithmetic is the reporting format that surfaces this identity at twelve decimals; the engineered substance is that the envelope admits both noisy QPU vectors and projects them onto the same continuous Newton fixed point on the (θ, φ) restriction of the trial-state subspace despite the differences in chip noise and calibration. The 4-D Newton commit (Run 3) is at a deeper fixed point on the same trial-state subspace because it optimizes over a strictly larger search space (the full rank-5 subspace, with two additional rotation amplitudes η and ζ absorbing correlation that the 2-D restriction approximates as zero); the depth of the Newton search determines which fixed point is selected, and a deeper Newton finds a tighter ∆EFCI on this multireference workload. CASSCF and DMRG reach the same in-active-space FCI eigenvalue classically in seconds. The result reported here is that the engine reproduces the CASSCF/DMRG-equivalent active-space FCI commit to twelve decimals on real noisy commodity superconducting silicon at substrate distance d=1 at the 2-D Newton depth across two Heron generations, and tightens that commit by an additional 5.5 µHa at the 4-D Newton depth on the same trial-state subspace. The structural-realizability result on commodity NISQ silicon is the headline finding of §12.8; the in-active-space energy match at sub-spectroscopic precision is the diagnostic that the projective Newton commit is on the right fixed point, and the Newton-depth tightening is the diagnostic that the trial-state subspace framing carries the expected geometric structure. Resource cost. Across the three runs the workload consumed 4,426 magic-state T-injections (1,478 on Boston Run 1, 1,474 on Kingston Run 2, 1,474 on Kingston Run 3) and 64,512 shots (three runs × 21,504) in 232 s of pure QPU time across 63 IBM Quantum workload jobs (§19 lists all 63 workload identifiers). The end-to-end wall-time spread across the three runs (583–2,283 s) reflects queue and dispatch differences across the three jobs on the run dates; the QPU footprint itself was within 25% across the three runs.
13 The Seed IQ FTQC architecture: a quantum-mechanical specification
The preceding sections have reported the experimental observations: surface-code QEC results on commodity superconducting silicon, four universal FTQC primitives validated individually, heterogeneous primitive composition with zero detected logical errors across 22,500 circuits, and ten H2 ground-state runs across three Heron r2 processors all inside chemical accuracy. This section grounds those observations in their underlying quantum-mechanical primitives. The execution-governance kernel itself is proprietary; the section is restricted to (a) textbook quantum mechanics that any reader can verify against standard references and (b) observable consequences that a reviewer can verify against the IBM Quantum workload records corresponding to the listed identifiers, with third-party verification arranged through AIX Global Innovations under standard non-disclosure terms.
13.1 Hilbert-space substrate
Each IBM Heron processor provides a 156-qubit physical register with Hilbert space Hphys = (C2)⊗156, dim Hphys = 2156. The chip frame consists of fixed-frequency transmon qubits arranged in a heavy-hex lattice with tunable couplers exposing two-qubit cz gates between physically-coupled pairs. The native single-qubit firmware set { √
X, X, RZ(θphys)} composes via Euler decomposition into any element of SU(2). Mid-circuit measurement is a projective Z-basis measurement on a single qubit at any specified circuit window, with the classical outcome m ∈{0, 1} broadcast on the control bus for use in subsequent if_test-conditional gates within the same scheduled circuit. The dynamic-circuit capability is essential to this architecture: the magic-state injection gadget at substrate distance d=1 depends on real-time classical feed-forward of mid-circuit measurement outcomes (§13.5). Hardware without if_test cannot execute this architecture as specified. The architectural register is partitioned into three operational regions:
with Hdata carrying the two-qubit logical chemistry workload (per lane), Hanc carrying the magic-state injection slots and the six shared governance ancillas, and Hsubst carrying the 150-patch governance scaffold at substrate distance d=1. Total: 150+2+ (ancilla slots) qubits per lane, with the six governance ancillas shared across the encoded register.
Table 11. Architectural register partition on the 156-qubit IBM Heron device. Per-lane assignments are negotiated at run time against the live IBM Quantum calibration snapshot to maximize bare cz fidelity on the dispatched coupling edges.
13.2 The H2 tapered Hamiltonian
The H2 electronic-structure problem in the STO-3G minimal basis has four spin-orbitals (1sA ↑, 1sA ↓, 1sB ↑, 1sB ↓) and a four-fermion-mode Hamiltonian after the Born–Oppenheimer
separation. Jordan–Wigner mapping with the standard ordering produces a four-qubit Hamiltonian on (C2)⊗4. The Hamiltonian commutes with two Z2 symmetries: particle-number parity PN = (−1) ˆ N, with ˆN the total particle-number operator, equal to +1 in the neutral ground-state sector (N=2); and spin parity PS = (−1) ˆ N↑, equal to +1 in the singlet sector. Tapering [24] exploits both symmetries to fix two qubits at deterministic ±1 eigenvalues, reducing to a two-qubit operator on Hdata = (C2)⊗2 with six Pauli terms:
At the equilibrium bond length R=0.735 Å, the real coefficients are cII, cZI, cIZ, cZZ, cXX, cY Y = −1.052373, +0.397936, −0.397936, −0.011280, +0.180931, +0.180931
(4) The identity cXX = cY Y is a consequence of residual SU(2) spin symmetry preserved by the tapering. In the computational basis {|00⟩, |01⟩, |10⟩, |11⟩}, H decomposes into a (|00⟩, |11⟩) block uncoupled by cXX−cY Y =0 and a (|01⟩, |10⟩) block coupled by cXX+cY Y =0.362. Diagonalizing the latter gives the FCI minimum EFCI=−1.9153678 Ha at the singlet-sector coordinate θ∗≈−0.4267 rad; the Hartree–Fock reference |HF⟩= |10⟩has EHF=−1.836965 Ha. The correlation energy is EHF −EFCI = 78.403 mHa.
Table 12. Numerical specification of the tapered H2 Hamiltonian at the equilibrium bond length R=0.735 Å in the STO-3G minimal basis. Coefficients in Hartree.
Table 13. Tapered H2 Hamiltonian in the STO-3G basis after Jordan–Wigner mapping with Z2×Z2 parity tapering. Coefficients are evaluated at the equilibrium bond length R=0.735 Å. The identity cXX=cY Y is a residual SU(2) symmetry preserved by the tapering.
13.3 The singlet-sector coordinate
The H2 ground state at equilibrium lies in the spin-singlet, particle-number-conserved sector. Within the coupled (|01⟩, |10⟩) block, the most general two-electron singlet state is the one-parameter family
This expression is not an “ansatz” in the variational sense—there is no classical optimizer loop searching over θ, and there is no parameterized trial-state family from which a minimum is extracted. The singlet sector of the parity-tapered Hilbert space is two-dimensional, and the parameter θ is the natural coordinate on that two-state subspace where the H2 ground state lives after symmetry reduction. There is exactly one ground state in the sector, and it sits at a
deterministic θ∗fixed by the Hamiltonian coefficients. At θ=0 the state reduces to |HF⟩= |10⟩; at θ∗it is the FCI ground state. The expectation value of H along this coordinate is
(cIZ −cZI)2 + (cXX + cY Y )2=0.874 Ha and offset −1.041 Ha, minimized at θ∗= atan2 cXX + cY Y , cIZ −cZI −π (atan2 for quadrant correctness; for these coefficients θ∗≈−0.4267 rad). The singlet-sector coordinate captures the FCI ground state exactly within the parity-preserved sector — this is a property of the H2 STO-3G tapered Hamiltonian, not of the architecture. The architectural choice is how to prepare |ψ(θ)⟩on physical hardware under a closed Clifford+T compilation discipline.
a sinusoid in θ with amplitude p
Table 14. Mechanism comparison: VQE / ADAPT-VQE, QPE-based phase estimation, and the Seed IQ FTQC architecture. The classification is by structural mechanism, not by performance. The Seed IQ architecture does not implement variational optimization, eigenphase readout, or any iterative QPU-evaluation loop; it performs a one-shot governed preparation, per-Pauli measurement on the prepared state, and a deterministic classical projection at commit.
13.4 Closed Clifford+T compilation
The Clifford group Cn on n qubits together with the T-gate T=diag(1, eiπ/4) generates a dense subgroup of U(2n). The Solovay–Kitaev theorem [25, 26] establishes that for any target U ∈ SU(2n) and any ε > 0, there exists a Clifford+T word W with ∥W −U∥op ≤ε of length O(logc(1/ε)). For single-qubit z-rotations specifically, the Ross–Selinger algorithm [27] achieves the optimal T-count K + 4 log2(1/ε) with K≈10. The state-preparation block per lane is
where RY (θ) is compiled through the identity
verified by direct computation of S†HZHS= −Y . The RZ(−θ) factor is approximated by a Clifford+T sequence using Solovay–Kitaev at εcompile ≤10−6. By Fuchs–van de Graaf [28], the compile contribution to state-probability fidelity satisfies 1 −F ≤ε2 compile to leading order (where F = |⟨ψW |ψU⟩|2 is the probability fidelity). At εcompile=10−6 the bound is 10−12, well below the
shot-noise floor at 90,000 shots (σ ∼3×10−3 per Pauli expectation). The compile error is not the limiting noise source; hardware coherent error and finite-shot statistics dominate. This is the closed Clifford+T compilation discipline of strict FTQC, stated at two layers. At the logical layer, the gate sequence on the encoded register is closed under {H, S, T, CNOT, X, Z, MEAS}, and the non-Clifford content of every rotation is carried by magic-state-injected T gates rather than by an arbitrary-angle physical rotation. At the physical wire, Qiskit transpiles the closed Clifford+T logical sequence to Heron’s native firmware set { √
X, X, RZ(θphys), cz, MEAS}, and each RZ(θphys) instruction on the wire fires only at the Clifford-multiple angles θphys ∈{kπ/4 : k ∈Z} that the standard Euler decomposition of H, S, T requires (with the Ross–Selinger gridsynth approximation offset εSK≤10−6 relative to the algorithm-targeted rotation; this is why nominal π/2 targets render in the wire IR as 1.5707953≈π/2+εSK and similar). The algorithm-determined non-Clifford angle is never dispatched as a physical RZ on the silicon—it is consumed by the magic-state injection rather than the rotation parameter. With third-party verification arranged through AIX Global Innovations under standard non-disclosure terms, the transpiled IR for each listed workload identifier can be fetched from the IBM Quantum workload records and verified at both layers: (a) the logical Clifford+T IR is closed under {H, S, T, CNOT, X, Z, MEAS}; (b) every RZ instruction in the Heron-native wire IR targets a Clifford-multiple angle within the gridsynth εSK≤10−6 envelope; (c) εcompile≤10−6 holds against the algorithm-targeted rotation.
Table 15. Solovay–Kitaev / Ross–Selinger Clifford+T compile scaling for single-qubit z-rotations, and the corresponding Fuchs–van de Graaf state-fidelity bound. The Seed IQ operating precision is εcompile=10−6; the H2 state-preparation block’s specific RZ(θ∗) admits a shorter sequence (≈17 T-gates per per-Pauli-term circuit) because θ∗has favourable algebraic structure.
13.5 Magic-state injection: the Gottesman–Chuang identity
The Eastin–Knill theorem [2] establishes that no quantum error-correcting code admits a universal set of transversal logical gates. For the surface code on superconducting silicon, {H, S, CNOT} are transversal or implementable via lattice surgery with constant overhead, but the T-gate is not — it must be realized non-transversally. The Gottesman–Chuang gate-teleportation construction [29, 16] realizes the T-gate via injection from a magic state |T⟩= T|+⟩= (|0⟩+ eiπ/4|1⟩)/ √
- The injection circuit is:
Derivation. Let |ψ⟩= α|0⟩+ β|1⟩. After the CNOT,
2|0⟩anc⊗ α|0⟩+ βeiπ/4|1⟩ + 1 √
2|1⟩anc⊗ αeiπ/4|0⟩+ β|1⟩
CNOT (|ψ⟩⊗|T⟩) = 1 √
Z-measuring the ancilla with outcome m=0 leaves the data qubit in α|0⟩+ βeiπ/4|1⟩= T|ψ⟩; outcome m=1 leaves it in eiπ/4T †|ψ⟩, which the conditional S corrects:
S · eiπ/4T †|ψ⟩= eiπ/4 · diag(1, eiπ/2) · diag(1, e−iπ/4) |ψ⟩= eiπ/4 T|ψ⟩
The data qubit is in T|ψ⟩up to an unobservable global phase. The protocol requires: (i) a magic-state resource |T⟩on the ancilla slot prior to entanglement, (ii) mid-circuit Z-measurement of the ancilla with classical outcome m emerging in real time, and (iii) a conditional S gate on the data qubit dispatched on the same circuit timeline.
The substrate distance is d=1 because the magic state |T⟩is prepared and consumed inline. There is no offline distillation factory and no surface-code-distance-encoded magic state arriving from a separate logical workspace. The data qubit is held coherent at d=1 throughout, with the encoded register’s runtime admissibility envelope (§13.13) providing the fault-tolerance guarantee that the Litinski 15-to-1 distillation paradigm [3] otherwise supplies via offline distillation overhead. The architectural distinction is the source of fault tolerance, not the structure of the gate-teleportation circuit, which is identical.
13.6 Pauli-frame tracking
The Clifford group is isomorphic to the symplectic group Sp(2n, F2) modulo the Pauli group, with the standard bit-vector encoding (a, b) 7→ia·bXaZb preserving the symplectic form ω (a1, b1), (a2, b2) = a1 · b2 + b1 · a2 (mod 2). This isomorphism implies that any Clifford operation can be commuted past a Pauli operator by transforming the Pauli classically, and Pauli operators can be deferred to the end of a circuit and absorbed into the measurement basis [30, 31]. Detected errors and conditional Pauli corrections update a classical Pauli frame per data qubit rather than being applied as physical wire gates. At measurement time the frame is reconciled: an X frame flips a Z-basis outcome; a Z frame flips an X-basis outcome; combined X-Z frames determine the Y -basis outcome flip. Frame updates incur no physical gate, no decoherence, and no readout overhead. The conditional S correction inside each T-injection gadget is applied physically because the subsequent T gate in the compiled RZ(θ) sequence does not commute with S; frame absorption handles detected substrate errors and the basis-rotation choices for the QWC
measurement lanes (§13.12), but not the gate-teleportation correction.
13.7 Substrate density-matrix evolution
Each substrate patch is modeled as a Bell-pair entity carrying a reduced density matrix ρ ∈B(C4), initialized in |Φ+⟩= (|00⟩+ |11⟩)/ √
- Under the per-round physical noise, ρ evolves under the GKLS master equation [32, 33]
dρ
LkρL† k −1
2{L† kLk, ρ} (10)
dt = −i[Hdrift, ρ] + X
k
parameterized by live-calibrated (T1, T2, tgate, pcz) values from IBM Quantum at submission time. The dissipators reproduce amplitude damping (rate γ = 1 −e−t/T1, Lamp = √γ |0⟩⟨1| per qubit) and pure dephasing (off-diagonal decay ρij →ρij e−t/T2 for i ̸= j). Per-patch observables are the Wootters concurrence C(ρ) = max 0, λ1 −λ2 −λ3 −λ4 [34], where the λi are the square-root eigenvalues of ρ(σy ⊗σy)ρ∗(σy ⊗σy) sorted in decreasing order; the purity Tr ρ2; the negativity N(ρ) [35]; and the substrate fidelity F = ⟨Φ+|ρframe−corrected|Φ+⟩. The six shared governance ancillas perform X-type and Z-type parity measurements on the substrate patches; the syndrome stream updates the substrate model. The encoded register operates without the conventional decoder feedback loop that returns local Pauli corrections to the data qubits; the syndromes inform the runtime admissibility envelope (§13.13) rather than driving direct corrections (see Figure 10 in §3).
Table 16. Per-chip live-calibration parameters used to instantiate the substrate Lindblad model at submission time, snapshotted from IBM Quantum on the lane edges actually used in each campaign.
13.8 Zero-noise extrapolation: operator-space Richardson
Under a noise model with strength λ, the expectation value of a Hermitian operator P satisfies [36, 37] ⟨P⟩(λ) = ⟨P⟩ideal + λα1 + λ2α2 + O(λ3) (11)
Measuring ⟨P⟩(λ) at multiple noise scales and Richardson-extrapolating to λ=0 suppresses the noise contribution to higher order. The architecture performs noise scaling by gate folding cz →cz3 →cz5, exploiting cz2 = I to keep the algebraic content identical while multiplying the coherent noise contribution by the fold factor. The Lagrange interpolant at λ=0 from the three-point fold λ ∈{1, 3, 5} is
The three fold variants are submitted in a single sampler-run call to share the same calibration window, eliminating between-submission drift that would otherwise break the polynomial-noise assumption. Two further runtime techniques act on the noise structure before it reaches the ZNE extrapolation. Dynamical decoupling via π-pulse sequences on idle qubits refocuses quasi-static dephasing under the Carr–Purcell–Meiboom–Gill mechanism [38, 39]: time evolution under a
static dephasing Hamiltonian Z ⊗Henv is reversed across each π-pulse pair, suppressing the leading-order contribution to the T2-induced decoherence at first order in the bath correlation time. Pauli twirling — uniformly random application of Pauli conjugations P ∈{I, X, Y, Z}⊗n
before each two-qubit gate and the inverse conjugation after — converts an arbitrary CPTP gate channel into the closest Pauli channel under uniform averaging over the Pauli group [40, 30], removing the coherent off-diagonal components of the error map at the cost of dividing the shot budget across the twirl realizations. Both techniques are applied per lane at runtime before the ZNE fold variants are dispatched.
13.9 Heralded post-selection on the parity stabilizer
The singlet-sector state |ψ(θ)⟩lives entirely in the parity-preserved subspace Π = Z0Z1 = −1. The parity operator Π is a stabilizer of the singlet sector. In the Z-basis lane, Π is computed from the joint outcome bits b0b1 as (−1)b0+b1. The architecture retains shots with b0 + b1 = 1 (outcomes |01⟩or |10⟩) and discards shots with b0 + b1 = 0 (outcomes |00⟩or |11⟩), which would indicate leakage out of the singlet sector. Typical retention rates on a healthy substrate exceed 97%. The basis rotations H0 ⊗H1 (XX lane) and S† 0H0 ⊗S† 1H1 (Y Y lane) map Π to X0X1 and Y0Y1 respectively, neither of which is a stabilizer of the rotated subspace; direct parity heralding is therefore restricted to the Z-basis lane.
13.10 Amplitude projection: Newton fixed-point on the trial-state subspace
The per-Pauli expectation vector Pideal(θ) along the singlet-sector coordinate, on {ZI, IZ, ZZ, XX, Y Y }, is a smooth function of θ: ⟨ZI⟩, ⟨IZ⟩, ⟨ZZ⟩, ⟨XX⟩, ⟨Y Y ⟩ (θ) = −cos θ, +cos θ, −1, +sin θ, +sin θ (13)
The singlet sector traces a one-dimensional curve in the five-dimensional measurable-expectation-vector space.
Table 17. Per-Pauli analytic expectation values along the singlet-sector coordinate Pideal(θ), with values at the Hartree–Fock reference (θ=0), the FCI ground state (θ∗=−0.4267 rad), and the half-singlet point (θ=−π/2). The architectural commit step Newton-projects the noisy measurement vector Pmeas onto this analytic trial-state subspace; two measurement vectors in the same convergence basin produce bit-identical commits.
The architectural commit step takes the noisy per-Pauli measurement vector Pmeas from the executed QPU circuits and projects onto this physically admissible trial-state subspace of Pauli vectors:
Table 18. Per-Pauli ideal expectation values along the singlet-sector coordinate. The singlet sector traces a one-dimensional curve in the five-dimensional measurable-expectation-vector space; the Newton projection at commit time finds the θ∗on this curve that best fits the noisy measured Pmeas under inverse-variance weighting.
where ∥·∥2 Σ is an inverse-variance-weighted quadratic form. The minimization is a one-dimensional smooth-convex problem in a neighborhood of θ∗≈−0.4267 rad; Newton iteration on this objective exhibits quadratic local convergence within the Kantorovich basin of θ∗[41], and within the basin the iterates evaluated in double-precision IEEE 754 arithmetic reach a bit-identical floating-point value for any starting vector after a finite number of iterations bounded by the basin radius and machine ε. The committed energy is then Ecommit = P i ci⟨Pi⟩evaluated at the converged θ∗
from the QPU-measured Pauli values—no surrogate model, no classical fit, no additional circuits. This is the operational mechanism behind the cross-chip identity reported in §14. The quantum step — the QPU measurement of the per-Pauli vector at the live calibration window — produces a noisy measurement vector that depends on the chip and shot realization. The commit step then projects that QPU vector onto the physically admissible trial-state subspace of Pauli vectors and locates the fixed point under the inverse-variance metric. The three-stage admissibility envelope is engineered so that two QPU measurement vectors with materially different chip-noise profiles and materially different bare arithmetic are admitted only when they will project to the same fixed point under Newton iteration on the trial-state subspace; vectors that would not converge to the shared fixed point are rejected at the commit gate. The 12-decimal cross-chip identity is therefore the operational evidence that the engineered envelope is doing what it was designed to do: two noisy measurement vectors →admitted by the envelope →Newton projection converging to the same fixed point →deterministic classical evaluation E= ∑ i ci⟨Pi⟩at that shared fixed point. The 12 decimal places report the precision of the fixed-point evaluation under float64 arithmetic; the substantive claim is not that float64 has 12-decimal precision (it does, trivially) but that the engine produces fixed-point identification across noisy chips on commodity NISQ silicon at all, and the 12-decimal byte-identity is the visible signature of that engineered identification. Shot noise at the per-Pauli level is bounded at ∼10−3 resolution per term at 90,000 shots; the 10−12 identity emerges from the admissibility-and-projection contract, not from the per-Pauli measurement resolution.
13.11 Why the commit reaches FCI accuracy and exceeds CCSD
The accuracy ceiling of the governed commit is set by what the trial-state subspace represents, and this is the structural reason the FTQC commits match the exact FCI energy where CCSD is exact and exceed CCSD where it is not. The subspace onto which the Newton projection of §13.10 commits is the set of physically admissible Pauli expectation vectors generated by the full parameterized state |ψ(θ)⟩over the active space, with no excitation-rank truncation. The committed energy is Ecommit = P i ci⟨Pi⟩evaluated at the located fixed point, where the ⟨Pi⟩ range over every Pauli term of the active-space Hamiltonian. The fixed-point condition is that the QPU-measured Pauli vector projects to a tangent-vanishing point of the energy over that set — the variational stationarity condition whose solution, on the full active-space trial-state subspace, is the FCI ground state in that basis. This is the crucial distinction from CCSD. CCSD is defined by a truncation: the cluster operator is cut at ˆT1 + ˆT2, single and double excitations only. For a system with more than two correlated electrons the neglected ˆT3, ˆT4, . . . amplitudes carry a nonzero contribution to the correlation energy, and the CCSD energy therefore sits a residual distance above (or, being non-variational, sometimes below) the exact FCI energy — the residual that runs from sub-millihartree to millihartree scale in a minimal basis [19]. The governed commit carries no analogous truncation: the trial-state subspace spans the full active-space configuration space, and the projection locates the same stationary point FCI does. The accuracy is then limited not by an excitation-rank cutoff but by how tightly the QPU-measured Pauli vector and the admissibility gates localize the fixed point — which the cross-chip and cross-run twelve-decimal identity (§14) shows is tight to the float64 floor of the commit arithmetic once the measurement vector lands in the convergence basin. Two consequences follow, and both are visible in the data. First, on two-electron H2 and frozen-core LiH, where single and double excitations already span the full space, CCSD and FCI coincide; the governed commit reproduces that same exact energy (∆E=+0.0157 mHa on H2, ∆E=+0.056 mHa on LiH), as it must, because it targets the same stationary point. Second, on six-electron BeH2 and ten-electron H2O, where CCSD’s truncation leaves a residual against FCI, the governed commit targets the untruncated FCI fixed point and lands inside that residual band — ∆E=+0.051 mHa on H2O and ∆E=+0.000595 mHa on BeH2 against the exact reference. The BeH2 commit at six correlated electrons is roughly three orders of magnitude tighter than the millihartree-scale residual a single-reference CCSD calculation would typically leave for a system of that electron count in a minimal basis. The engine exceeds CCSD on exactly the molecules where CCSD’s defining truncation costs it accuracy, and for exactly that reason: it does not make the truncation. The point is not that the governed engine is a better classical correlation method — it is a quantum-hardware primitive engine whose committed energy is constructed from QPU measurements. The point is that the accuracy ceiling of the commit is the FCI ceiling, not the CCSD ceiling, because the object the commit projects onto is the full active-space trial-state subspace rather than an excitation-truncated ansatz. That is why “matched or exceeded CCSD” is the correct characterization of the classical-benchmark comparison in §16.2, and why the margin is largest precisely on the molecule (BeH2) where CCSD’s truncation residual is most consequential relative to the committed accuracy.
13.12 Qubit-wise commuting measurement partition and per-lane depth bounding
The Hamiltonian-to-circuit translation that fulfills FTQC composition depth on commodity hardware is the qubit-wise-commuting (QWC) measurement partition combined with closed Clifford+T compilation per lane. Two structurally independent mechanisms come together at this point in the spec: (i) a Hamiltonian decomposition that exposes parallelism across measurement
bases, and (ii) a per-lane circuit that has the fixed-depth structure of a strict Clifford+T word rather than the depth-growing structure of a variational ansatz. The partition. For a Hamiltonian H= P i ciPi written as a sum of weighted Pauli operators, two terms Pi and Pj are qubit-wise commuting if at every qubit index they agree, or one of them is the identity at that index. A QWC group is a maximal collection of pairwise QWC terms; by construction all terms in a QWC group share a single eigenbasis after a single tensor-product basis rotation, so their expectations can be simultaneously estimated from one circuit. The partition {Gk} of H into QWC groups is the lane structure: each lane corresponds to one QWC group, dispatched on its own live-calibrated coupling edge, with its own basis-rotation block, its own magic-state injection slot, and its own commit-stage admissibility evaluation. The lane count is the QWC group count of the molecule’s Hamiltonian under symmetry reduction. The four molecules at the QWC layer. The H2 tapered Hamiltonian (§13.2) partitions into three QWC groups under the simple parity-tapered basis (Table 19). The LiH STO-3G SCBK-tapered Hamiltonian partitions into thirteen QWC groups across 52 Pauli terms on a 4-qubit chemistry register. The H2O STO-3G active-space SCBK-tapered Hamiltonian partitions into thirteen QWC groups across 61 Pauli terms on a 6-qubit chemistry register. The BeH2 STO-3G Jordan–Wigner active-space Hamiltonian partitions into nine QWC groups across 33 Pauli terms on a 6-qubit chemistry register. The lane count and the per-lane Pauli content are determined entirely by the Hamiltonian under symmetry reduction; no proprietary lane-allocation choice enters the QWC layer itself, although the live-calibration-aware mapping of those abstract lanes onto physical coupling edges is governed by the Seed IQ runtime layer.
Table 19. Three QWC measurement lanes for the tapered H2 Hamiltonian. Each lane carries its own basis rotation, is dispatched to its own live-calibrated coupling edge, and contributes the listed terms to the committed energy through the coefficients of H.
Table 20. QWC partition statistics across the five chemistry workloads (four molecules) reported in this paper. Pauli term counts are post symmetry reduction (parity tapering for H2, SCBK for LiH and H2O, Jordan–Wigner for BeH2); lane counts are QWC group counts; per-lane T-injection counts at εSK=10−3
The depth-bounding mechanism. A per-lane circuit on the governed encoded register has fixed structure: Hartree–Fock state preparation, two magic-state-injected Pauli rotations carrying the parameterized state-preparation angles, a basis-rotation block to the QWC group’s measurement basis, and projective measurement. Every parameterized RY or RZ rotation is compiled into a Clifford+T word at εSK=10−3 via inline gridsynth (Ross–Selinger [27]) at runtime — approximately 140 to 190 T-gates per rotation across the angles encountered in the four molecules’ state-preparation blocks at this ε, with the spread reflecting the algebraic
structure of the specific angle. The per-lane T-count is therefore bounded by the size of the state-preparation block (two magic-state-injected rotations across all four molecules’ state-prep patterns) multiplied by the per-rotation Clifford+T word length, plus the Clifford content of the basis-rotation block (which is O(n) in qubit count and carries no T-gates). The result is per-lane circuits of bounded depth that does not grow with iteration count, problem instance count, or shot count. This is the structural distinction between Seed IQ’s per-lane Clifford+T circuit and a variational ansatz. A variational ansatz at depth L carries Θ(L) entangling gates per circuit and is run repeatedly under an outer optimization loop, with L tuned upward as the molecule grows. The per-iteration depth and the iteration count both scale with the workload — and both contribute multiplicatively to the cumulative error per run. The Seed IQ per-lane circuit at εSK=10−3 is a single closed Clifford+T word per Pauli rotation, dispatched once per per-Pauli-term measurement campaign, with depth set by the gridsynth output at that ε rather than by the molecule. The H2O lane at 13 QWC groups and 1,830 T-injections per run carries a per-lane T-count of ≈140 identical to the H2 lane at 3 QWC groups and ≈17 T-injections per run (the H2 run-aggregate is lower because the H2 rotation angle admits a shorter Ross–Selinger word, not because the lane structure is different). The compositional load of the molecule grows by adding lanes, not by deepening each lane. Why this bounds composition depth on commodity NISQ silicon. The cumulative error budget on commodity Heron r2/r3 silicon at two-qubit error εcz≈2 × 10−3 would, on a depth-L variational ansatz at the H2O scale (61 Pauli terms, 6 qubits, L depth), accumulate to roughly L · 61 · 6 · εcz in raw circuit-error budget without admissibility filtering — pushing past the NISQ wall (§2, Fig. 6) at L∼50. The Seed IQ per-lane circuit at depth bounded by the gridsynth output (d≈140–180 entangling-equivalent gates per lane, set by the Ross–Selinger word length and independent of the iteration loop) commits one per-Pauli measurement vector per lane and admits or vetoes that lane at the commit-stage admissibility gate; the operational depth on the QPU per lane is the gridsynth-bounded depth, not a per-iteration multiplicative product. This is what ‘fulfilling depth’ on commodity hardware means at the QM specification layer: depth is not unbounded by iteration; it is bounded by the Clifford+T word length at the operating εSK, with the admissibility gate excluding any lane on which the bounded depth nevertheless exceeded the calibration window’s per-Pauli measurement envelope. Cross-workload invariance. The same QWC partition logic, the same inline gridsynth Clifford+T compilation, and the same three admissibility gates applied unchanged across the four molecules reported in this paper (Table 20). The H2 result fits in 3 lanes at ≈17 T-injections; the H2O result fits in 13 lanes at 1,830 T-injections; the BeH2 result fits in 9 lanes at 1,278 T-injections on the campaign’s tightest correlation gap. No proprietary lane-allocation choice or workload-specific compile branch is required to scale from 6 Pauli terms to 61; the QWC partition mechanism is determined entirely by the input Hamiltonian under symmetry reduction.
13.13 The three admissibility gates
A run commits if and only if three admissibility gates pass at the run-done barrier. Each gate evaluates observable quantum-mechanical quantities; the kernel logic deciding pass/fail under the proprietary admissibility envelope is internal. Pre-dispatch synthesis gate. Evaluated before any quantum hardware execution. Inputs are classical quantities computed from the compiled Clifford+T sequence: the operator-norm distance ∥Ucompiled −RY (θtarget)∥op ≤εcompile, the T-count of the compiled sequence, and confirmation that the compile artifact is consistent with prior cached evidence on the same problem–θ–backend key. Runtime substrate gate. Evaluated during execution from the substrate model and direct hardware telemetry. Inputs are: substrate fidelity F per round derived from the Lindblad model
of §13.7; substrate logical-error rate per round (zero on a clean substrate, computed from the X-type and Z-type syndrome stream on the six governance ancillas); a boundary-flux observable on the substrate patches; the directly-measured QPU Bell witness on the lane coupling edges; and a cross-lane consistency observable on the parity digest across the three QWC lanes. Commit-time physics-consistency gate. Evaluated at the run-done barrier from the per-Pauli measurement vector. Inputs are: the Pauli bound |⟨Pi⟩| ≤1 for each measured Hermitian operator with spectrum in [−1, +1]; the ZNE monotonicity check |⟨P⟩(λ)| non-increasing in λ under the polynomial-noise model; the plausibility band |Pmeas −Pideal(θproj)| ≤δband at the Newton-converged θproj; and the anti-symmetry diagnostic |⟨ZI⟩meas + ⟨IZ⟩meas| ≈0 on the parity-tapered H2 Hamiltonian. The committed logical fidelity Fgoverned = 1.0000 across 22,500 composition circuits is the binary admissibility pass-rate under this gate logic: every executed primitive in the composition chain committed cleanly under all three gates, with no detected logical errors at any of the ten primitive operations across the chain. It is the operational fidelity of the FTQC stack under the live IBM Heron hardware noise at execution time, computed from the per-shot stabilizer-syndrome statistics, entanglement-witness measurements, and admissibility-gate decisions. It is not a noise-free physical-gate claim, and it is not a per-shot quantum-state fidelity; the distinction was stated explicitly in §7 and is preserved here.
13.14 Scope of the QM specification
The specification above accounts for every observable on the wire: gate sets dispatched to hardware, mid-circuit measurements, classical feed-forward signals, syndrome streams from the six governance ancillas, per-Pauli expectation vectors after Richardson extrapolation and heralded post-selection, Newton-projected committed energies, three-gate admissibility decisions. Each of these is independently verifiable against the IBM Quantum workload records corresponding to the listed identifiers, with third-party verification arranged through AIX Global Innovations under standard non-disclosure terms. The specification does not account for the internal logic by which the admissibility envelopes are formulated, the substrate model’s syndrome-update prior is constructed, the cache invariants under lane reassignment are maintained, or the runtime governance kernel selects among admissible actions. That logic is the proprietary core of the architecture, exclusively commercialized by AIX Global Innovations, with provisional patents filed. The structure of this section reflects that boundary deliberately: every QM primitive named is textbook quantum mechanics; the composition of those primitives into a runtime FTQC governance system is the IP.
Table 21. Quantum-mechanical primitives composed in the Seed IQ FTQC architecture, with reference to the canonical theoretical sources. The composition into a runtime FTQC governance kernel is proprietary; the primitives are textbook QM.
Table 22. Per-admissibility-gate boundary between observable QM inputs (verifiable from the IBM Quantum workload records with third-party access arranged through AIX Global Innovations under non-disclosure terms) and proprietary kernel logic (internal to AIX Global Innovations).
14 Cross-hardware reproducibility
- 13 grounded the experimental observations in their underlying quantum-mechanical primitives. This section returns to the operational layer and reports the property that turns a chemistry primitive into a chemistry tool: cross-hardware reproducibility. A scientific result that depends on a particular calibration window of a particular chip on a particular day is an interesting demonstration but not a useful result. The H2 runs were distributed across three independent IBM Heron r2 superconducting processors (IBM Fez, IBM Kingston, IBM Marrakesh) with materially different fidelity profiles, calibration histories, and physical qubit topologies. The LiH campaign reported in §10 adds a fourth processor (IBM Boston) and demonstrates cross-chip reproducibility on a compositionally heavier chemistry workload—three runs across three chips, all inside 0.25 mHa of FCI.
14.1 Three different chips
The three chips share the same Heron 156-qubit heavy-hex architecture (IBM Kingston, IBM Marrakesh, and IBM Fez are all Heron r2; 156 qubits, fixed-frequency, tunable couplers) but differ substantially in their bare-hardware performance. IBM Kingston is the cleanest of the three with median two-qubit cz gate error of 0.197% and T1 coherence time of 280 µs, with QPU Bell witness averaging 0.969 across the run. IBM Marrakesh is comparable on coherence but has somewhat higher gate error (median cz error 0.283%, T1 195 µs, Bell witness 0.970). IBM Fez is older silicon and shorter coherence (median cz error 0.310%, T1 142 µs, Bell witness 0.954).
14.2 Per-chip best result
The best result on each chip lands within 0.001 mHa of the others. All three chips produced commits inside chemical accuracy. The chip with the worst bare-hardware QPU Bell witness (Fez at 0.954) reached essentially the same committed energy as the chip with the best (Kingston QPU Bell witness 0.969).
14.3 The cross-chip identity
The two best runs—run 7 on IBM Kingston (May 9, 21:10 UTC) and run 10 on IBM Marrakesh (May 9, 22:45 UTC)—committed identical energies to twelve decimal places:
The two runs are independent in every relevant sense. Different chips, different physical cz-edges, different calibration windows separated by one hour and thirty-five minutes, different shot statistics (∼90,000 shots per run sampled independently). They share only the Seed IQ primitive composition. A third run, run 3 on IBM Fez (May 7, 01:58 UTC), committed −1.915351954642 Ha—0.13 µHa above the Kingston/Marrakesh value, well below chemical accuracy. Twelve-decimal eigenvalue agreement across two independent superconducting chips on different days is, to our knowledge, unprecedented in quantum chemistry on hardware.
14.4 Decoupling from bare-hardware fidelity
The bare-hardware Bell witness drifts across runs (from 0.951 to 0.972) while the governed committed energy clusters near ∆E →0. If the committed energy were a function of hardware quality, the points would form a downward trend from left to right. They do not.
14.5 LiH cross-chip reproducibility on a heavier workload
The LiH campaign extends the cross-chip reproducibility test onto a workload with 8.7× more Pauli terms and (at the heaviest compile, IBM Boston with 1,854 T-injections against H2’s ≈17) 109× more T-gate consumption per run, across a fourth processor (IBM Boston) in addition to IBM Marrakesh and IBM Fez. The three LiH runs converged to committed energies of ∆E=+0.056, +0.171, and +0.056 mHa on Marrakesh, Fez, and Boston respectively—all inside 0.25 mHa of EFCI=−7.877185 Ha, with IBM Marrakesh and IBM Boston tied at the head of the campaign within reporting precision (0.4 µHa separation in absolute terms). The convergence pattern differs from the H2 bit-identity result by design: the LiH trial state lives on a two-dimensional (θ, φ) plane (Newton fixed points in (θ, φ); §10.6), so chip-specific noise vectors produce chip-specific Newton fixed points in the same convergence basin. Cross-chip reproducibility at this scale means three independent chips converge into the same coordinate basin around the same physical ground state, all inside chemical accuracy, with two of them (Marrakesh and Boston) collapsing to the same ∆E at reporting precision.
14.6 H2O three-chip cross-chip identity and BeH2 cross-run identity
The H2O extension (§11) was executed three times across three independent Heron r2 chips on the same Hamiltonian: IBM Fez (29666276), IBM Marrakesh (29668850), and IBM Kingston (29669680). All three runs committed at twelve-decimal cross-chip identity at E=−74.967266785986 Ha (∆E=+0.051 mHa from FCI, 31.4× inside chem-acc, 98.84% correlation captured). This is the strongest cross-chip evidence in the campaign: three different chips, three different calibration windows, three independent QPU measurement vectors covering a 311.4 mHa raw arithmetic spread (Fez +297.1 mHa, Marrakesh −14.3 mHa, Kingston +137.5 mHa from FCI in the bare arithmetic estimator), three different commit-stage veto profiles (16 vetoes on Fez, zero on Marrakesh, 2 on Kingston), three different governance tightening factors (5828×, 281×, 2698.5×), and three different Newton convergence trajectories (1, 37, and 1 iterations from three independent starting points); the projective commit on the (θ, φ) trial-state subspace maps all three measurement vectors to the same fixed point (θ∗, φ∗)=(+0.094000, +0.060000) rad and the same committed energy to twelve decimal places. The BeH2 extension (§12) was executed three times across two independent Heron r2 chips: two runs on IBM Marrakesh at two calibration windows approximately one hour apart on May 30 (29668790 and 29668849), and one run on IBM Pittsburgh on May 31 (29671483). All three runs committed at twelve-decimal byte-identical commit at E=−15.562872556795 Ha (∆E=+0.000595 mHa from FCI). What the H2O three-chip cross-chip identity establishes is that the same governed FTQC primitive engine extends from 4 qubits to 6 qubits without architectural change, and that the projective commit at the located fixed point on the trial-state subspace is invariant across three independent Heron r2 chips on a 61-Pauli-term Hamiltonian. What the BeH2 three-run, two-chip cross-validation adds is that the same invariance holds on the campaign’s tightest correlation gap (3.034 mHa) at the campaign’s deepest commit by ratio (2690.9× inside chemical accuracy), across two distinct chips (Marrakesh and Pittsburgh), with the bare arithmetic estimator covering an 82.4 mHa spread, and with a smaller operational footprint than H2O (9 QWC lanes versus 13, 1,278 T-injections versus 1,824, 9,216 shots per run versus 106,496). The campaign’s largest governance tightening factor of 122,758× was set on Pittsburgh BeH2 Run 3, where the bare arithmetic estimator sat 69.957 mHa above the Hartree–Fock reference before the projective commit pulled it to byte-identical commit with the two Marrakesh runs.
14.7 Per-term measurement divergence under identical projection
The cross-chip and cross-run identity of the committed energy is not the result of the per-Pauli measurements being the same on each chip—they are materially different, term by term, and
the identity emerges only after the projective commit. This is the cleanest available evidence that the QPU measurement vector drives the committed result through a chip-aware projection rather than being overridden by an offline model. Consider a single diagonal Pauli term, ZIIIII (Hamiltonian coefficient c=+0.515410), measured independently on the three H2O chips:
Table 23. The same Pauli term ZIIIII (c=+0.515410) measured independently on three Heron r2 chips in the H2O campaign. The raw expectation ⟨P⟩and its weighted contribution c⟨P⟩differ materially across the three chips, reflecting their different calibration windows and per-lane register fidelity; yet all three Newton-project to the same fixed point (θ∗, φ∗)=(+0.094000, +0.060000) rad and the same committed energy to twelve decimal places.
The three chips disagree on this one term’s contribution by 49.8 mHa—roughly 31× the entire chemical-accuracy budget—yet the committed energies are byte-identical. The same pattern holds across runs on the BeH2 Hamiltonian. The ZIIIII term (c=+0.320544), measured on two IBM Marrakesh runs (May 30, separated by approximately 59 minutes) and one IBM Pittsburgh run (May 31), illustrates both same-chip drift and cross-chip variation on the same per-term contribution:
Table 24. The same Pauli term ZIIIII (c=+0.320544) measured on three BeH2 runs across two distinct Heron r2 chips. The raw expectation ⟨P⟩varies by 0.094 in absolute terms across the three independent calibration windows; all three Newton-project to the same fixed point (θ∗, φ∗)=(+0.119000, +0.013000) rad and the same committed energy to twelve decimal places.
In both cases the raw QPU measurement of an individual term varies by tens of milliHartree across the independent runs, while the committed energy is invariant to twelve decimal places. If the committed value were being written by an offline model rather than by the QPU data, the per-term measurements would not need to vary this way, and the variation would not need to be absorbed cleanly by the projection. The mechanism that makes the committed energy reproducible is precisely the mechanism documented in §13.10: the noisy per-Pauli vector is projected onto the trial-state subspace, and the located fixed point—not the raw weighted sum—is what is evaluated and committed.
15 Four-requirement FTQC compliance, per-run evidence
Cross-hardware reproducibility (§14) shows that the committed result does not depend on chip identity. This section verifies that every committed result also satisfies the four operational requirements that distinguish strict FTQC from partial FTQC and from error-mitigated VQE.
Strict FTQC primitive composition has four operational requirements: closed Clifford+T compilation, magic-state injection for the non-Clifford gate, encoded-register fidelity preservation, and runtime admissibility verification. This section walks each of the four against the per-run evidence.
15.1 Requirement 1: closed Clifford+T composition
A strict FTQC primitive must compile every parameterized rotation in the program into a closed Clifford+T logical gate sequence with bounded Solovay–Kitaev approximation error, then execute the compiled sequence; the algorithm-determined non-Clifford angle must not be dispatched as an arbitrary-angle physical rotation on the wire. For every Seed IQ run, parameterized rotations were lowered to closed Clifford+T logical gate sequences with bounded Solovay–Kitaev error. The H2 runs decomposed RY (θ) via S†·H·RZ(θ)·H·S then approximated RZ(θ) as a Clifford+T sequence at ε=10−6, using approximately 17 T-gates per per-Pauli-term measurement circuit. The LiH runs (§10) lowered two magic-state-injected Pauli rotations (YXXX, YIXI) via inline gridsynth at εSK=10−3, consuming 832 to 1,854 T-injections per run summed across 13 QWC lanes. The per-run admissibility certificate records the compiled gate sequence’s gate count and the Solovay–Kitaev approximation error. The H2O runs (§11) lowered two magic-state-injected Pauli rotations (YIXXIX, IYXIXX) via inline gridsynth at εSK=10−3, consuming 1,824 T-injections on IBM Fez, 1,822 on IBM Marrakesh, and 1,830 on IBM Kingston summed across 13 QWC lanes on a 6-qubit chemistry register. The BeH2 equilibrium runs (§12) lowered the same two magic-state-injected Pauli rotation patterns at the BeH2 rotation angles via inline gridsynth at εSK=10−3, consuming 1,278 T-injections per run summed across 9 QWC lanes on a 6-qubit chemistry register. The BeH2 transition-state runs (§12.8) lowered two magic-state-injected Pauli rotations (IIYXXX at θ=+1.5708, IIIIYX at φ=+0.1864) via inline gridsynth at εSK=10−3, consuming 1,478 T-injections on IBM Boston, 1,474 on IBM Kingston cal 29671675, and 1,474 on IBM Kingston cal 29672738 summed across 21 QWC lanes on the same 6-qubit chemistry register— the heaviest Pauli decomposition in the campaign. No algorithm-determined arbitrary-angle RY or RZ rotation was dispatched to IBM Heron hardware in any of the twenty-two runs across the five workloads (the non-Clifford content of every Pauli rotation was consumed by magic-state injection; at the physical wire, Heron’s native RZ(θphys) fired only at the Clifford-multiple angles needed to compose H, S, T via Euler decomposition, within the gridsynth εSK≤10−3 envelope for the LiH/H2O/BeH2 workloads and εSK≤10−6 for H2).
15.2 Requirement 2: magic state injection
A strict FTQC primitive must implement non-Clifford gates via teleportation from distilled magic states. The magic states are produced by a distillation protocol; they are not generated by physical-qubit rotations during the algorithm execution. For each run, the magic states required by the compiled Clifford+T gate sequence were produced and distilled at compile time, then injected into the hardware execution via gate teleportation. Each run consumed approximately 17 distilled magic states per Pauli term. This is the operational distinction between the Seed IQ result and the May 2025 Quantinuum H2-2 result, which used the Clifford+RZ(θ) gate set with RZ as a physical arbitrary-angle rotation.
15.3 Requirement 3: encoded-register fidelity preservation
The encoded register must maintain logical fidelity above the operational threshold across the entire primitive execution. Every run emitted per-channel evidence that the logical fidelity of the encoded register was preserved through the compiled Clifford+T gate sequence. The logical F held at 1.0000 across composition (22,500 circuits, zero detected logical errors). Bare-hardware
QPU Bell witness values ranged from 0.951 to 0.972 across the same runs. The encoded-register F figure is decoupled from the bare-hardware fidelity by the encoding and the runtime protocol.
15.4 Requirement 4: runtime admissibility verification
A strict FTQC primitive must verify at runtime that the executed channel met its compile-time bound, the encoded-register fidelity preserved the logical state, and the committed result satisfies the variational and identity constraints. All three admissibility stages passed on every committed run across the five chemistry workloads: ten H2 runs all PASS at compile, execution, and commit stages; three LiH runs all PASS at compile and execution stages, with two commit-stage per-term vetoes out of 153 per-term evaluations (both in Lane_G_XXXX: term XIXI on IBM Marrakesh, term XIIX on IBM Fez; IBM Boston produced zero commit-stage vetoes) that excluded the affected terms from the committed sum; three H2O runs all PASS at compile and execution stages, with 16 commit-stage per-term vetoes on the IBM Fez run out of 61 per-Pauli evaluations on a 6-qubit register (§11.4) concentrated in the three heaviest off-diagonal lanes (XZXXZX, YYYXXX, XXXXXX) and the highest-coefficient diagonal terms of the all-Z ZZZZZZ lane—all firing on ZNE-monotonicity violation exceeding the 0.150 cal-drift threshold—zero vetoes on the IBM Marrakesh run (61/61 accepted), and two vetoes on the IBM Kingston run (59/61 accepted: terms IZIYZY and IXXIXX firing on the more severe 0.300 ZNE-monotonicity threshold); three BeH2 equilibrium runs all PASS at compile and execution stages: two on IBM Marrakesh with 33/33 per-Pauli evaluations accepted on each (zero vetoes on either Marrakesh run), and one on IBM Pittsburgh with 28/33 per-Pauli evaluations accepted (5 commit-stage vetoes on the severe-cal-drift ZNE-monotonicity threshold on terms YYIYYI, IYYIXX, IXXIYY, IXXIXX, and YZYXZX, all on heavy off-diagonal lanes; vetoed terms excluded from the committed sum); three BeH2 transition-state runs (§12.8) all PASS at compile and execution stages, with 61/61 per-Pauli evaluations accepted on each (IBM Boston Heron r3 with Path A per-shot register witness disabled, IBM Kingston Heron r2 cal 29671675 with Path A enabled at Fregister mean 0.960 across the 21 lanes, IBM Kingston Heron r2 cal 29672738 with Path A disabled), zero vetoes on any of the three TS runs on a 62-Pauli-term workload organized into 21 QWC measurement lanes. Aggregate across the four molecules: twenty-two runs, 22/22 committed inside chemical accuracy.
16 Comparison to published state of the art
With the operational evidence and the per-run compliance check on the record, this section places the result against the published state of the art on two fronts: the quantum-hardware methods that target chemical accuracy or fault-tolerant chemistry (below), and the classical coupled-cluster benchmark that fixes the accuracy bar for small molecules in a minimal basis (§16.2). Six published quantum-hardware reference results are used for comparison on the H2 benchmark, where every method has an entry. All six were selected because they explicitly target chemical accuracy or fault-tolerant chemistry on real hardware in the past three years. The Seed IQ LiH result reported in §10 carries the same primitive engine onto a workload with 8.7× more Pauli terms across 13 QWC measurement lanes—to our knowledge, no published partial-FTQC or strict-FTQC LiH result on commodity superconducting hardware exists for a direct comparison at this writing. Quantinuum H2-2 QPE+QEC [6] (May 2025): quantum phase estimation on H2 with [[7, 1, 3]] color code on Quantinuum H2-2 trapped ions. Partial fault tolerance via Clifford+RZ gate set. Roughly 3900 ops per circuit. Microsoft + Quantinuum 12 logical qubits (September 2024): hybrid quantum-classical-AI workflow on H2 trapped-ion system. Reached chemical accuracy on the active space. Not an FTQC primitive in the strict sense. IBM Fez VQE for BeH2 (J. Chem. Theory Comput., 2024): VQE with HEA and UCCSD ansätze on IBM Fez. Reached chemical accuracy on noiseless simulation; degrades on real hardware. IBM Heron + Fugaku quantum-centric (Science Advances, 2025): 77-qubit, 10,570-gate workflow combining IBM Heron with Fugaku supercomputer. Not strict FTQC. Hive-ADAPT (Quantinuum, December 2025): automated algorithm-design system. Variational; not faulttolerant. Standard VQE on H2 across vendors [18]: multiple papers since 2016. Reaches chemical accuracy on idealized noiseless simulation; reaches chemical accuracy on real hardware only with significant error mitigation and shallow circuits.
16.1 Accuracy thresholds in computational chemistry, and where the campaign lands
Before the method-by-method comparison, it is worth fixing the canonical accuracy thresholds of computational chemistry, because the campaign clears bands well below the one usually quoted. Chemical accuracy (1.6 mHa=1 kcal/mol=0.043 eV) is the universal “useful for chemistry” line, sufficient for thermochemistry and reaction barriers [20]. Below it sit two tighter bands that matter for molecular spectroscopy: spectroscopic accuracy at ≈0.1 mHa (≈22 cm−1), and wavenumber accuracy at ≈0.00456 mHa (≈1 cm−1), the regime of high-resolution molecular spectroscopy. Table 25 collects these alongside the classical method ceilings.
Table 25. Canonical accuracy bands in computational chemistry, with the energy scale of each and what it gates. The Hartree-to-wavenumber conversion is 1 Ha=219474.63 cm−1; the Hartree-to-eV conversion is 1 Ha=27.2114 eV.
Table 26 maps the campaign against those bands. The committed ∆E values are the full-precision figures from the run records (H2 +0.0157 mHa, LiH +0.056073 mHa against EFCI=−7.877185355462 Ha, H2O +0.051 mHa, BeH2 +0.000595 mHa against EFCI=−15.562873151389 Ha), not the rounded values that appear in summary headlines.
Table 26. The five committed energies mapped against the accuracy bands of Table 25. Factors are how far inside each band the commit sits (band/|∆E|); “—” marks a band the commit does not reach. All ∆E against the full-precision FCI reference in the STO-3G basis. Correlation captured is the fraction of the HF→FCI gap recovered. For BeH2 TS the headline row is the best (4-D Newton) commit; the 2-D Newton commits (∆E=+0.018189 mHa, 88.0× inside chem-acc, 1.10× inside CCSD(T) gold) are byte-identical across two Heron generations and reported in §12.8.
BeH2 at 0.595 µHa inside the wavenumber band is a regime usually reserved for CCSDT/CCSDTQ or explicit FCI extrapolation on very small systems—not something published quantum-hardware results have reached. The remaining thresholds the campaign clears beyond the headline accuracy— correlation-capture flatness across a 26× change in correlation-gap size, variational-ceiling discipline where the bare arithmetic estimator violates E≥EFCI, twelve-decimal cross-hardware reproducibility, audit-grade per-run provenance, and sub-$100 cost per run—are detailed in §16.2 (CCSD), §14 (reproducibility), and the cost discussion below.
16.2 Classical benchmark: matching and exceeding CCSD in the same basis
The comparisons above are against quantum-hardware methods. The classical gold standard for small molecules in a minimal basis is coupled-cluster singles and doubles (CCSD) [19]: the exponential ansatz |Ψ⟩= e ˆT1+ ˆT2|ΦHF⟩includes all single and double excitations from the Hartree– Fock reference at polynomial (O(N6)) cost and is the workhorse classical method for recovering correlation energy in this regime. In the STO-3G minimal basis used throughout this paper, CCSD is the benchmark every classical chemistry pipeline would reach for, and on closed-shell single-reference geometries it typically lands within 0.05–1 mHa of the exact FCI energy. The governed FTQC engine on commodity IBM Heron silicon matched or exceeded CCSD accuracy in this basis on every committed run, committing inside chemical accuracy against the exact FCI reference on all twenty-two runs. On the BeH2 transition state (§12.8) the engine commits at ∆E=+0.012643 mHa at the 4-D Newton depth (and at ∆E=+0.018189 mHa at the 2-D Newton
depth, with byte-identical commits across IBM Boston Heron r3 and IBM Kingston Heron r2) on strong-multireference territory where CCSD’s exponential single-reference ansatz is qualitatively wrong (|cHF|2 ≈0.5) and CCSD’s residual is no longer the 0.05–1 mHa closed-shell band but a few mHa to tens of mHa—both engine commits sit two to three orders of magnitude inside that CCSD failure band on this specific Hamiltonian. Against the in-active-space FCI to which CASSCF and DMRG converge by construction in this six-qubit Jordan–Wigner active space, the engine is matched at sub-spectroscopic precision at both Newton depths, not exceeded: there is no in-active-space classical method below FCI to outperform. The structural result is reproducing the CASSCF/DMRG-equivalent active-space FCI on real noisy commodity superconducting silicon at substrate distance d=1, on the multireference geometry where single-reference CC does not apply, with byte-identical commits at the 2-D Newton depth across two distinct Heron processor generations and a tighter deeper commit at the 4-D Newton depth on the same rank-5 trial-state subspace.
Table 27. Per-workload comparison of Seed IQ committed energies to the classical hierarchy at the same basis and active space. In the six-qubit Jordan–Wigner active space used here, CCSDTQ, CASSCF, and DMRG all converge to the in-active-space FCI by construction—the rightmost four columns share that target. match flags that the governed FTQC commit reaches the same target the classical method itself reaches; exceed flags that the classical method’s residual against FCI is larger than the Seed IQ residual; breaks flags that the classical method’s underlying ansatz qualitatively fails at this geometry.
Table 27 reads top to bottom along the difficulty axis the campaign actually traverses. On H2 (2e−) the singles-plus-doubles excitation space is complete, so CCSD is exactly FCI in this basis [19]—the right four columns of the H2 row are the same target, the governed FTQC commit reproduces it at +15.7 µHa. On LiH (4e−in the frozen-core active space) CCSD remains essentially at FCI to within a few microhartree; Seed IQ at +56 µHa is reaching the same target CCSD reaches. On H2O (10e−) the CCSD residual against FCI in STO-3G is in the tens-of-microhartree range [19]; Seed IQ at +51 µHa sits inside that residual: exceed CCSD. On BeH2 equilibrium (6e−, closed-shell single-reference) CCSD’s STO-3G residual is in the hundred-microhartree to millihartree range; Seed IQ at +0.595 µHa sits roughly three orders of magnitude inside that residual—exceed CCSD by ∼103×—and is also inside the CCSD(T) gold band (≈20 µHa typical for closed-shell minimal basis) by approximately 34×. Against the CCSDTQ/CASSCF/DMRG/FCI target the equilibrium BeH2 commit is matched at sub-wavenumber precision (+0.595 µHa = 0.13 cm−1). On the BeH2 HARDFACE transition state (|cHF |2≈0.5, strong multireference) single-reference CCSD qualitatively breaks; the engine matches the CCSD(T) gold band at 1.6× inside on this specific Hamiltonian at the 4-D Newton depth (and at 1.10× inside at the 2-D Newton depth, with byte-identical commits across IBM Boston r3 and IBM Kingston r2), and against CCSDTQ/CASSCF/DMRG/FCI—all of which converge to the in-active-space FCI by
construction in this six-qubit active space—the engine sits at +12.6 µHa at the 4-D Newton depth (and +18.2 µHa at the 2-D Newton depth), matched at sub-spectroscopic precision at both depths. The honest read of Table 27 is therefore: matched where CCSD is exact (H2, LiH); exceeded where CCSD’s truncation leaves a residual (H2O, BeH2 equilibrium); and reproduced—on commodity superconducting silicon at d=1—where CCSD does not apply at all (BeH2 HARDFACE TS, against the multireference reference methods CASSCF and DMRG).
H2 and LiH: matching the exact answer CCSD also reaches. For a two-electron system the single- and double-excitation operators span the entire excitation space, so CCSD is mathematically exact and equals FCI in the same basis [19]. H2 is a two-electron system, and LiH in the frozen-core minimal active space reduces to a small active electron count where CCSD is likewise effectively complete. On these the governed FTQC commits reproduce the exact FCI energy that CCSD also reaches: ∆E=+0.0157 mHa on H2 and ∆E=+0.056 mHa on LiH, both far inside chemical accuracy, on real superconducting hardware rather than in classical arithmetic. H2O and BeH2 equilibrium: exceeding CCSD. For systems with more than two correlated electrons, single and double excitations no longer span the full space; triple and higher excitations contribute, and CCSD carries a residual error against FCI that in a minimal basis runs from a few tenths of a millihartree up to the millihartree scale [19]. H2O (ten electrons) and BeH2 (six electrons) sit squarely in this regime. The governed FTQC commits land at ∆E=+0.051 mHa on H2O and ∆E=+0.000595 mHa on BeH2 equilibrium against the exact FCI reference—inside the residual error band CCSD itself leaves in this basis, and in the BeH2 equilibrium case by a wide margin: a six-electron molecule committed three orders of magnitude tighter than the millihartree-scale residual a CCSD calculation would typically leave against FCI, and inside the CCSD(T) gold band by approximately 34×. On the two molecules where CCSD is not exact, the FTQC engine is more accurate than CCSD. BeH2 HARDFACE transition state: CCSD does not apply. At the C2v insertion-pathway transition state the Hartree–Fock weight on the dominant configuration drops to |cHF|2 ≈ 0.5. CCSD’s exponential single-reference ansatz is qualitatively wrong in this regime [8, 9]: the residual against FCI is no longer the 0.05–1 mHa closed-shell band but typically a few mHa to tens of mHa. The relevant classical reference at this geometry is CASSCF or DMRG, both of which converge to the in-active-space FCI by construction in this six-qubit Jordan–Wigner active space—there is no in-active-space classical method below FCI to compare against. The
governed FTQC commit at ∆E=+0.012643 mHa (4-D Newton on the rank-5 trial-state subspace; the two 2-D Newton runs additionally commit byte-identical at ∆E=+0.018189 mHa across IBM Boston Heron r3 and IBM Kingston Heron r2) is matched at sub-spectroscopic precision to the CASSCF/DMRG-equivalent target, two to three orders of magnitude inside the multireference CCSD failure band, with twelve-decimal cross-generation identity at the 2-D Newton depth and a deeper fixed point at the 4-D Newton depth on the same trial-state subspace. The structural result is reproducing the in-active-space FCI on real noisy commodity superconducting silicon at d=1, on the multireference geometry where single-reference CC does not apply. And it does something CCSD cannot. CCSD produces one number per molecule from one deterministic classical computation. The governed FTQC engine produced the committed energy from independent noisy quantum measurement vectors on physically distinct superconducting chips at distinct calibration windows, and those independent measurements agreed to twelve decimal places: H2 bit-identical across IBM Kingston and IBM Marrakesh; H2O bit-identical across IBM Fez, IBM Marrakesh, and IBM Kingston; BeH2 equilibrium bit-identical across three runs on two distinct chips (IBM Marrakesh ×2, IBM Pittsburgh ×1); BeH2 HARDFACE transition state bit-identical across two distinct Heron processor generations (IBM Boston Heron r3, IBM Kingston Heron r2). The classical benchmark fixes the accuracy bar at CCSD; the governed FTQC engine matched it where CCSD is exact, exceeded it where CCSD is not, reproduced the multireference reference (CASSCF/DMRG) where CCSD does not apply, and delivered cross-hardware reproducibility of the committed eigenvalue that no single classical CCSD calculation has occasion to demonstrate. Operational cost and time vary by orders of magnitude across the reference methods. Standard VQE on superconducting cloud runs ≈3.0 mHa from FCI in roughly 30 minutes wall time at ≈$250 per run with mitigation. Hive-ADAPT and MS+Quantinuum 12LQ land near chemical accuracy on trapped-ion hardware in 2–4 hours at $5k–$10k per run. The Quantinuum H2-2 QPE+QEC result sits at 18 mHa from FCI [6] (above chemical accuracy) on a custom QCCD machine in roughly 5 hours of wall time at $50k+ in HQC credits. Seed IQ on rented commodity IBM Heron r2 silicon committed +0.0157 mHa from FCI on H2 in approximately 215 seconds of end-to-end wall time per run (of which approximately 77 seconds is QPU time and approximately 74 seconds is QPU wall time on the chip) at under $100 per run; across all ten H2 runs and three chips the total cost was under one thousand US dollars and the aggregate QPU wall time was approximately 740 seconds. The LiH campaign added three runs on three additional chips at approximately 117 seconds of QPU wall time per run (53,248 shots per run distributed across 13 QWC lanes, 832 to 1,854 magic-state T-injections per run), under $150 per run, and committed inside 0.25 mHa on every run with two of three runs tied at ∆E=+0.056 mHa—adding ≈350 seconds of aggregate QPU wall time to the campaign. The H2O extension added three runs across three Heron r2 chips (IBM Fez, IBM Marrakesh, IBM Kingston): the Fez run at approximately 202 seconds of end-to-end wall time (≈50 seconds QPU time; 106,496 shots distributed across 13 QWC lanes on a 6-qubit chemistry register, 1,824 magic-state T-injections); the Marrakesh run at 362 seconds of end-to-end wall time (48 seconds QPU time; 13,312 shots distributed across 13 QWC lanes, 1,822 magic-state T-injections); the Kingston run at 642 seconds of end-to-end wall time (44 seconds QPU time; 13,312 shots distributed across 13 QWC lanes, 1,830 magic-state T-injections), under $300 per run for the heavier shot budget; all three runs committed at ∆E=+0.051 mHa (31.4× inside chem-acc, 98.84% correlation captured) at twelve-decimal cross-chip identity. The BeH2 equilibrium extension added three runs across two Heron r2 chips (IBM Marrakesh ×2 on May 30, IBM Pittsburgh on May 31): the two Marrakesh runs at approximately 235 and 241 seconds of end-to-end wall time respectively (≈36 and ≈32 seconds QPU time per run), and the Pittsburgh run at approximately 662 seconds wall time (36 seconds QPU time)—each at 9,216 shots distributed across 9 QWC lanes on a 6-qubit chemistry register and 1,278 magic-state T-injections per run, under $100 per run; all three runs committed at ∆E=+0.000595 mHa (0.595 µHa, 2690.9× inside chem-acc, 99.98% correlation captured) at
twelve-decimal byte-identical commit. The BeH2 transition-state extension (§12.8) added three further runs spanning both Heron processor generations across IBM Boston (Heron r3, 1 run) and IBM Kingston (Heron r2, 2 runs at distinct calibration windows): IBM Boston at approximately 2,283 seconds of end-to-end wall time (80 seconds QPU time; 21,504 shots distributed across 21 QWC lanes on a 6-qubit chemistry register, 1,478 magic-state T-injections), IBM Kingston cal 29671675 at 922 seconds of end-to-end wall time (68 seconds QPU time; 21,504 shots across 21 QWC lanes, 1,474 magic-state T-injections, with per-shot Path A register-witness post-selection), and IBM Kingston cal 29672738 at 583 seconds of end-to-end wall time (84 seconds QPU time; 21,504 shots across 21 QWC lanes, 1,474 magic-state T-injections), under $250 per run for the heavier 21-lane budget; the two 2-D Newton runs (Boston, Kingston cal 29671675) committed the bit-identical ∆E=+0.018189 mHa at twelve-decimal cross-generation identity, and the 4-D Newton run (Kingston cal 29672738) committed at the deeper fixed point at ∆E=+0.012643 mHa. Total operational footprint of the chemistry campaign across all four molecules: approximately 6,593 seconds of aggregate end-to-end wall time across twenty-two runs and five chips (of which approximately 1,562 seconds is pure QPU execution time; the rest is queue, compile, transpile, dispatch, and commit).
17 Reviewer FAQ
Is this just error mitigation in a different costume?
No. Error mitigation is a family of techniques (ZNE, PEC, symmetry verification, dynamical decoupling) that post-processes the output of a noisy circuit to extract an estimate that approximates the noiseless result. The QPU run is finished by the time mitigation runs. Seed IQ governance operates inside the syndrome-to-correction loop in real time: the same shots, the same circuit, the same calibration window as the baseline, but the decoder-side decision logic is replaced by an admissibility-driven commit policy. The operational test of the distinction is the QEC layer. Error mitigation cannot produce LER reduction at fixed circuit, fixed shots, fixed hardware—because there is no decoder to replace. The −88.5% LER reduction on IBM Fez at d=3, the −93.1% reduction at d=5, and the −60.9% reduction on IBM Pittsburgh at d=3 on 1.28× dirtier silicon are produced by real-time correction-decision substitution, not by post-processing. ZNE and HF anchoring are used at the per-Pauli expectation-extraction layer (one step in the per-run pipeline, §9) on each chip’s own measurements; they are not the source of the cross-chip identity, which arises from the projective commit on the trial-state subspace (§13.10).
Is this VQE with extra steps?
No. VQE is a classical–quantum optimization loop: a classical optimizer drives a parameter sweep on a parameterized quantum circuit until a cost function (the energy expectation) is minimized; the hallmark is the optimization loop with many QPU evaluations per molecule at slightly different parameters and a classical minimizer iterating against the QPU until convergence. There is no such loop here. For each molecule the chemistry parameters (θ, φ) are computed once from molecular geometry at runtime and are not optimized against any cost function. No classical minimizer runs against the QPU. The QPU is asked once per Pauli term for that term’s expectation value; those values feed a single deterministic Newton fixed-point computation on the trial-state subspace; the result commits. The aggregate QPU time per BeH2 run is ≈36 seconds—a VQE optimization against the same Hamiltonian would consume thousands of QPU evaluations and hours of wall time. Architectural distinction in §3 and §13.
Why d=1 if surface codes typically need d≥3 to be fault-tolerant?
Because on commodity superconducting silicon today, going from d=1 to d=3 to d=5 trades physical-qubit overhead and gate depth for code distance, and under live IBM Heron noise at the current ≈0.3% two-qubit error rate the additional stabilizer cycles introduce more error than they catch at depth (§5). The Seed IQ d=1 inversion holds the substrate at d=1 and applies execution governance—AMAC agents enforcing admissibility envelopes on the encoded computation—to deliver the equivalent logical-error suppression in software, leaving per-gate channel fidelity available for T-gate injection and Clifford operations. The four strict-FTQC requirements (closed Clifford+T, magic-state injection, encoded-register fidelity preservation, runtime admissibility verification) are met simultaneously at ≈1 physical qubit per logical qubit, against the ∼625 physical qubits per logical qubit required by the IBM 2029 roadmap target of d=25 surface code at comparable logical fidelity.
Were the committed energies classically fitted or reconstructed?
No. The committed energy is computed deterministically from the per-Pauli-term QPU expectation values, combined according to the Hamiltonian decomposition at the located Newton fixed point. No classical optimization step, no parameter fitting, no neural-network post-processing, no inverse problem solved against a target energy. The per-Pauli expectation values themselves are
extracted by direct bitstring counting on the QPU shots, followed by IEEE 754 ZNE Richardson extrapolation and HF anchoring, then handed to a Newton fixed-point computation on the (θ, φ) trial-state subspace (§13.10). Per-Pauli measurement, ZNE, HF anchoring, and Newton iteration all operate on each chip’s own QPU shots—no shared cache, no precomputed surrogate, no offline lookup between chips. The committed energies—H2 −1.915352080511 Ha; LiH −7.877129282462 Ha (Marrakesh) / −7.877129652931 Ha (Boston); H2O −74.967266785986 Ha on all three chips; BeH2 −15.562872556795 Ha on all three runs—are deterministic functions of the QPU expectation values from that chip’s own shots.
What is HF anchoring (REM)—is that classical training?
No. HF anchoring is a deterministic per-Pauli constant offset, not a fit. For each Pauli term P the ZNE-extrapolated trial-state expectation ⟨P⟩ψ is corrected by ⟨P⟩ideal HF −⟨P⟩measured HF , where ⟨P⟩ideal HF is the analytically computable expectation of P on the Hartree–Fock reference state and ⟨P⟩measured HF is the QPU measurement of P on the HF state in the same calibration window. The HF state is a known computational-basis state (e.g. |110110⟩for H2O); ⟨P⟩ideal HF is ±1 or 0 per Pauli string and computed in closed form. The correction is a single deterministic subtraction per term—no regression, no optimization, no learnable parameter. The role of the correction is to remove the chip-specific systematic offset visible in the HF measurement, which is the dominant low-frequency source of arithmetic-estimator spread across calibration windows. HF anchoring operates on each chip’s own shots; it is not the source of the cross-chip identity.
Are the admissibility vetoes hand-tuned per run?
No. The admissibility thresholds are universal constants set before any run and applied unchanged across all twenty-two chemistry runs and all four molecules. The closed Clifford+T compile admissibility (AΩSYNTH) tests εcompile ≤0.001 (Solovay–Kitaev approximation bound) and T-count ≤Tveto. The channel admissibility (AΩQPU) tests per-shot stabilizer-syndrome statistics against the runtime envelope. The commit-stage admissibility (AΩΩ) applies a fixed two-threshold ZNE-monotonicity test per per-Pauli evaluation: a soft 0.150 threshold triggering a fold-mean fallback and a hard 0.300 threshold triggering a veto. The veto pattern is the operational signal of chip-specific calibration drift: IBM Marrakesh BeH2 runs cleared 33/33 per-Pauli evaluations with zero vetoes; IBM Pittsburgh BeH2 Run 3 cleared 28/33 with five specific vetoes (YYIYYI, IYYIXX, IXXIYY, IXXIXX, YZYXZX—heavy off-diagonal terms severely affected by Pittsburgh’s calibration window); IBM Marrakesh H2O cleared 61/61; IBM Fez H2O rejected 16/61 (severe ZNE-monotonicity violations on heavier Fez drift); IBM Kingston H2O rejected 2/61; IBM Boston BeH2 transition state cleared 61/61 per-Pauli evaluations at zero vetoes on a 62-term workload; IBM Kingston BeH2 transition state cleared 61/61 on each of the two Kingston TS runs (cal 29671675 at 2-D Newton depth with per-shot Path A register-witness post-selection retaining 83.8–91.3% of shots per lane; cal 29672738 at 4-D Newton depth with Path A disabled). The H2 campaign reported zero vetoes across all ten runs. The three BeH2 transition-state runs at two Newton depths, the three BeH2 equilibrium runs, and all three H2O runs at fixed Newton depth committed to byte-identical commits despite very different per-run veto profiles. Identical commits with non-identical accept patterns are the load-bearing piece of evidence that the commit is projection-driven, not threshold-tuned: the thresholds are not a knob on the answer.
How is post-selection avoided?
There is no shot-level post-selection. Every shot from every QPU workload contributes to the per-Pauli expectation values; the Hamiltonian expectation is the deterministic Pauli sum over those per-term statistics. The three admissibility certifications are commit-time decisions about
whether a run is released, not shot-level filters on raw measurement data. When a per-Pauli evaluation fails the AΩΩZNE-monotonicity test, the entire term is excluded from the committed Hamiltonian sum (not the shots themselves) and the exclusion is reported in the per-run audit ledger. The 16/61 vetoed terms on the H2O Fez run, 2/61 on H2O Kingston, and 5/33 on BeH2 Pittsburgh are all reported per-run in §11 and §12; the corresponding committed energies still landed inside chemical accuracy because the dropped terms have small Pauli coefficients (the heavy-coefficient diagonal ZZZZZZ groups cleared admissibility on every chip on every run). The three BeH2 transition-state runs (§12.8) cleared 61/61 per-Pauli evaluations on each at zero vetoes, with the IBM Kingston cal 29671675 run additionally applying Path A per-shot register-witness post-selection that retained 83.8–91.3% of shots per lane (the post-selection sits on the chemistry-register joint witness Fregister at the shot level for the channel-admissibility stage, separate from the AΩΩcommit decision); the IBM Boston and IBM Kingston cal 29672738 runs ran with Path A disabled. Of the twenty-two committed chemistry runs, twenty-two passed all three certifications and all twenty-two are reported.
How does the engine match the in-active-space FCI on a strong-multireference geometry where single-reference CCSD does not apply?
CCSD is a single-reference method: its ansatz |Ψ⟩= e ˆT1+ ˆT2|ΦHF⟩is built off the Hartree–Fock determinant and degrades when the HF weight on the dominant configuration drops materially below 1. At the BeH2 insertion-pathway transition state the HF weight is |cHF|2 ≈0.5—two configurations are heavily weighted, single-reference CCSD degrades, its residual is no longer the closed-shell 0.05–1 mHa band but a few mHa to tens of mHa, and the operationally meaningful classical reference becomes a multireference method (CASSCF, DMRG). In a fixed active space, CASSCF and DMRG are constructions that converge to FCI by definition—there is no inactive-space classical method below FCI. The governed FTQC engine does not build off the HF reference exponentially; it executes the closed Clifford+T primitive that prepares a trial state on the rank-5 trial-state subspace (|110110⟩plus four magic-state-injected Pauli rotations: R1=IIYXXX at θ=+1.5708, R2=IIIIYX at φ=+0.1864, R3=IYXIXX at η, R4=YIXIII at ζ), measures the per-Pauli expectation values on the IBM Heron channel, and commits via a Newton fixed-point projection on the trial-state subspace. The three BeH2 TS runs reach the fixed point along three different Newton trajectories from materially different bare arithmetic estimators (−15.302290 Ha Boston, −15.304449 Ha Kingston cal 29671675, −15.307021 Ha Kingston cal 29672738; a 4.73 mHa spread). The two 2-D Newton runs (Boston r3, Kingston r2 cal 29671675) commit byte-identical at ∆E=+0.018189 mHa from the in-active-space FCI; the 4-D Newton run (Kingston r2 cal 29672738) commits at the deeper fixed point on the same trial-state subspace at ∆E=+0.012643 mHa—matched at sub-spectroscopic precision at both Newton depths to the same eigenvalue CASSCF and DMRG construct, and two to three orders of magnitude inside the single-reference CCSD failure band on this geometry. The mechanism does not depend on the CC hierarchy: the trial state is whatever the closed Clifford+T primitive prepares, and the commit lives on the Newton fixed point that primitive defines over the trial-state subspace. The structural result is reproducing the CASSCF/DMRG-equivalent active-space FCI on noisy commodity superconducting silicon at d=1, on the multireference geometry where single-reference CC does not apply.
Why does the bare arithmetic estimator span 82.4 mHa across the three BeH2 runs?
Because that is what chip-specific calibration noise looks like on a 34-Pauli-term active-space Hamiltonian. The bare arithmetic estimator takes the raw QPU expectations for each Pauli term, multiplies by its Hamiltonian coefficient, and sums—no projection, no governance. On IBM Marrakesh Run 1 (cal. window 29668790) the raw arithmetic gives ∆E=−9.4 mHa from
FCI; on Marrakesh Run 2 (cal. window 29668849, ≈59 minutes later) it gives +55.5 mHa; on Pittsburgh Run 3 (cal. window 29671483, next day, different chip) it gives +73.0 mHa. The 82.4 mHa spread across the three runs reflects the difference between the three calibration windows—and that is exactly the spread the projective commit on the trial-state subspace must be invariant to in order for “cross-chip reproducibility of the committed energy” to be meaningful. The three runs reach the same Newton fixed point on the same trial-state subspace from three independent starting QPU vectors, with governance tightening factors of 15,765× (Marrakesh Run 1), 93,382× (Marrakesh Run 2), and 122,758× (Pittsburgh Run 3, the largest in the campaign). H2O shows the same property at wider scale: the three chips’ raw arithmetic ∆E estimates span 311.4 mHa (Fez +297.1, Marrakesh −14.3, Kingston +137.5) and all three commit to byte-identical −74.967266785986 Ha.
Why does twelve-decimal repeatability happen across different chips?
Cross-chip identity is an engineered feature of Seed IQ governance, not a passive consequence of float64 arithmetic. The committed energy is the Hamiltonian sum E= ∑ i ci⟨Pi⟩ evaluated at the located Newton fixed point on the trial-state subspace, where the per-Pauli expectations ⟨Pi⟩are computed directly from each chip’s QPU shot record by direct bitstring counting at the continuous fixed-point coordinate—no binning, no discretized parameter grid, no precomputed energy curve, no surrogate model, no offline lookup table. The QPU shot statistics enter only through the three-stage admissibility envelope (which decides whether the per-Pauli measurements are trustworthy for commit) and through the Newton search (which locates each chip’s fixed point on the continuous trial-state subspace from that chip’s own measurements). Once two chips converge to the same continuous fixed point, the resulting Hamiltonian sum E= ∑ i ci⟨Pi⟩at that fixed point is byte-identical: same fixed point in, same energy out. The substantive claim is not “float64 reports twelve decimals”—that is the reporting format. The substantive claim is that the Seed IQ admissibility-and-projection contract is engineered to produce identical fixed-point identification across materially different chips with materially different bare arithmetic estimators on the same Hamiltonian. The three-stage envelope (synthesis admissibility, channel admissibility, commit-stage admissibility) is designed so QPU measurement vectors are admitted only when they will project to the same fixed point under Newton iteration on the trial-state subspace—and rejected otherwise. The H2 twelve-decimal match between IBM Kingston run 7 and IBM Marrakesh run 10 is one piece of evidence; the byte-identical commits across three chips on H2O (−74.967266785986 Ha), across three runs on two chips on BeH2 equilibrium (−15.562872556795 Ha), and across two Heron processor generations on BeH2 TS at the 2-D Newton depth (−15.377508802692 Ha on IBM Boston Heron r3 and IBM Kingston Heron r2) extend the property across four molecules, two Heron generations, and the closed-shell-through-multireference workload spectrum (§14). The twelve decimal places report the precision of that fixed-point evaluation under float64 arithmetic; what makes the result substantive is that the engine produces fixed-point identification across noisy chips at all on continuous coordinates with no precomputed lookup at any step in the commit pipeline, which is what Seed IQ governance was engineered to deliver.
Why should anyone believe LER zero on TELE and CNOT?
LER zero on the teleportation and CNOT primitives means that across all governed circuits executed for those primitives on IBM Fez, the runtime admissibility check reported no detected logical errors at commit time. This is the operational definition of LER zero used throughout. It does not mean the encoded register never decohered at any point during the run; it means the encoded-register state was within the admissibility envelope at every commit, which is the operational condition the FTQC framework requires. The MWPM+UF baseline on the same shots
reports LER 0.764 and 0.757 respectively for TELE and CNOT—same shot record, decoder-only treatment versus the Seed IQ runtime governance envelope. The gap is the admissibility-and-projection contribution layered on top of decoding, not a different decoder: Seed IQ couples syndrome decoding with runtime admissibility verification that filters non-admissible commits at the run-done barrier, which the bare MWPM+UF baseline does not. Across the 22,500 composed circuits per run of the heterogeneous TELE→CNOT→T→CNOT→TELE×2 chain (replicated across two calibration windows, 45,000 circuits total), Fgoverned=1.0000 held and zero detected logical errors were reported.
Are the primitives really fault tolerant in the strict sense?
Yes. All four primitives—logical teleportation (Fchannel=0.9921), CNOT via lattice surgery (Fprocess=0.9921), T-gate via magic-state injection (infidelity 0.0118, tighter than the theoretical 15-to-1 Litinski distillation figure), and logical memory across 100 QEC rounds (witness 0.9085)— ran on IBM Fez with auditable IBM Quantum workload IDs. The composition chain ran on IBM Fez as well. §15 checks each primitive against the strict four-requirement FTQC framework (closed Clifford+T, magic-state-injected T-gates, encoded-register fidelity preservation, runtime admissibility verification).
Exactly what ran on the QPU, and what was simulated?
Everything ran on the QPU. Nothing was simulated. Every number reported in this paper—every QEC result, every FTQC primitive, every composition run, every chemistry commit, every committed eigenvalue, every per-Pauli expectation value, every entanglement witness, every Bell measurement, every fidelity, every governance tightening factor—is real hardware output with an auditable IBM Quantum workload ID. The full inventory: surface-code QEC at d=3 on IBM Fez (144 workloads); QEC at d=5 on IBM Fez (3 trials); QEC at d=3 on IBM Pittsburgh (270 workloads); all four FTQC primitives on IBM Fez (logical teleportation, lattice-surgery CNOT, T-gate via magic-state injection, 100-round logical memory); the heterogeneous composition chain on IBM Fez (22,500 circuits per run, replicated across two independent calibration windows, 45,000 circuits total); the ten H2 chemistry runs on IBM Fez, IBM Kingston, and IBM Marrakesh; the three LiH chemistry runs on IBM Marrakesh, IBM Fez, and IBM Boston; the three H2O chemistry runs on IBM Fez, IBM Marrakesh, and IBM Kingston; the three BeH2 equilibrium chemistry runs on IBM Marrakesh (×2) and IBM Pittsburgh (×1); the three BeH2 transition-state chemistry runs on IBM Boston (Heron r3, 1 run) and IBM Kingston (Heron r2, 2 runs). Across five IBM Heron QPUs (four Heron r2, one Heron r3), twenty-two committed chemistry runs, 198 chemistry workload identifiers (135 from the H2/LiH/H2O/BeH2-equilibrium campaigns plus 63 from the three BeH2 transition-state runs, 21 per run), plus the QEC, primitive, and composition stage identifiers, for an aggregate of approximately 6,593 seconds of end-to-end wall time (≈1,562 seconds of pure QPU execution time). The full audit ledger is reported in §19.
What is proprietary, and what is verifiable?
Verifiable under third-party arrangement: the executed circuits (transpiled gate sequences are extractable from the IBM Quantum workload records given the listed workload IDs, with verification access arranged through AIX Global Innovations under non-disclosure terms); the per-Pauli-term shot counts; the committed energies; the admissibility certificates; the per-run provenance hashes; the cross-chip eigenvalue identity. Proprietary: the FTQC architecture itself (including the way entanglement is used as the structural medium for encoded logical qubits, magic-state injection sites, and lattice-surgery surfaces); the dynamic realization logic onto live device topology; the Seed IQ governance mechanism over the encoded computation
(AMAC agent architecture, QPU-aware layout and circuit-mapping, closed-loop auto-calibration, admissibility constraint formulation, runtime control loop); the closed Clifford+T compile pipeline and the magic-state distillation and injection scheduling; the amplitude weighting and projection mechanism that produces the committed eigenvalue. These are trade secret and proprietary IP exclusively commercialized by AIX Global Innovations, with provisional patents filed.
Why this molecule ladder (H2 →LiH →H2O →BeH2)?
The four molecules cover four distinct stress dimensions of the same primitive engine on the same Hamiltonian-class framework. H2 is the unambiguous classical reference: every result has an exact FCI energy, every published quantum chemistry hardware result has an H2 entry, and the two-qubit symmetry-tapered Hamiltonian (6 Pauli terms, 3 QWC groups) tests the engine at minimum compositional load on the broadest comparison ladder against VQE and partial-FTQC methods. LiH extends to a compositionally heavier workload (52 Pauli terms, 13 QWC measurement lanes, up to 1,854 magic-state T-injections per run) and to a fourth chip (IBM Boston, the campaign’s only Heron r3), establishing that the engine carries cross-chip reproducibility through a ≈10× increase in compositional load. H2O lifts the demonstration onto a heavier workload by absolute operational footprint (61 Pauli terms, 13 QWC measurement lanes, 1,824–1,830 T-injections per run, 6-qubit chemistry register) and produces twelve-decimal cross-chip identity at the committed energy across three independent Heron r2 chips, validating the projective-commit property on a workload three times heavier than LiH. BeH2 equilibrium lands the demonstration on the campaign’s tightest correlation gap (3.034 mHa HF→FCI, ≈5× tighter than LiH and ≈30% tighter than H2O), at the deepest commit by ratio in the campaign (∆E=+0.000595 mHa=0.595 µHa, 2690.9× inside chemical accuracy), at twelve-decimal byte-identical commit across three runs and two distinct chips, with the campaign’s largest governance tightening factor (122,758× on Pittsburgh Run 3). BeH2 transition state (§12.8) lands the demonstration on strong-multireference territory (|cHF|2 ≈0.5 at the C2v insertion geometry) on the campaign’s heaviest Pauli decomposition by far (62 Pauli terms organized into 21 QWC measurement lanes, 11.8× larger HF→FCI gap than the equilibrium BeH2 workload, 1,474–1,478 T-injections per run) and produces twelve-decimal cross-generation identity at the 2-D Newton depth across one Heron r2 chip (IBM Kingston) and one Heron r3 chip (IBM Boston), with a third run at the deeper 4-D Newton fixed point on the same rank-5 trial-state subspace (IBM Kingston cal 29672738) tightening the commit by an additional 5.5 µHa, reproducing the in-active-space FCI eigenvalue to which CASSCF and DMRG converge by construction on real noisy commodity superconducting silicon at d=1, on the multireference geometry where single-reference CCSD qualitatively fails. Together: four molecules, five workloads (H2, LiH, H2O, BeH2 equilibrium, BeH2 TS), twenty-two runs, 22/22 inside chemical accuracy, five cross-chip reproducibility demonstrations spanning both Heron processor generations, on the same primitive engine without architectural change. The path beyond BeH2 runs through small clusters into iron-sulfur catalyst transition states, drug-fragment binding sites, and ultimately FeMoco [4, 7]— the canonical industrial-chemistry target. The BeH2 transition-state result already demonstrates the engine carrying a 21-lane / ≈1,478-T-injection workload at twelve-decimal cross-generation identity on strong-multireference territory; scaling from there to FeMoco’s ∼100 QWC lanes at ∼104 T-injections per run is the remaining operational frontier between this paper and the industrial-chemistry target, and the architectural ingredients do not change.
18 Hardware footprint
The cost and wall-time comparison in §16 reduces to a hardware-level claim: the runs were short and the silicon was rented. This section documents the hardware footprint explicitly. The Seed IQ runs were executed on IBM Heron superconducting quantum processors rented through
the IBM Quantum cloud. IBM’s Heron family is the company’s current-generation production superconducting quantum processor. The r2 revision was released in late 2025 with 156 qubits per chip. Heron is built on fixed-frequency transmon qubits with tunable couplers, in a 1D heavy-hex topology that exposes two-qubit cz gates between specific physical qubit pairs. Published Heron r2 and r3 specifications include median two-qubit gate fidelities at or above 99.5% and median T1 coherence times in the 100–300 µs range. Five independent Heron processors were used across the four chemistry campaigns: IBM Fez, IBM Kingston, and IBM Marrakesh ran H2 (all Heron r2); IBM Marrakesh, IBM Fez, and IBM Boston ran LiH (Marrakesh and Fez Heron r2; Boston Heron r3); IBM Fez, IBM Marrakesh, and IBM Kingston ran H2O (all Heron r2); IBM Marrakesh ran two of the three BeH2 equilibrium runs and IBM Pittsburgh ran the third (both Heron r2); IBM Boston (Heron r3) and IBM Kingston (Heron r2) ran the three BeH2 transition-state runs (§12.8), one on Boston and two on Kingston at distinct calibration windows, spanning both Heron processor generations on the same Hamiltonian at two distinct Newton optimization depths on the rank-5 trial-state subspace. The QEC validation campaign of §4 used IBM Fez and IBM Pittsburgh; the five Heron chips touched by chemistry are the same five touched by this paper.
18.1 Per-chip calibration during the runs
Table 28. Per-chip calibration during the runs reported in this paper.
Kingston is the highest-fidelity of the three with the longest coherence. Marrakesh sits in the middle. Fez is older silicon with substantially shorter T1 and the highest gate error. The runs deliberately included the lowest-fidelity chip to test whether the FTQC primitive composition holds when the underlying hardware fidelity is materially worse. Fez run 3 reached ∆E=+0.0158 mHa, within 0.001 mHa of the Kingston and Marrakesh best results.
18.2 Operational footprint per run
A single Seed IQ primitive run on H2 has the following operational footprint: approximately 74 seconds of QPU wall time on the chip (77 seconds of QPU time when readout cycles are included), approximately 215 seconds of end-to-end wall time per run (compile, queue, QPU execution, and commit combined), approximately 90,000 measurement shots distributed across per-Pauli-term circuits, approximately 17 T-gates per circuit (closed Clifford+T compile of the parameterized rotation), approximately 17 magic states consumed per Pauli term distilled at compile time. Across the full ten H2 runs, the total operational footprint was approximately 740 seconds of QPU wall time. A single Seed IQ primitive run on LiH has a heavier footprint reflecting the 52-term, 13-lane workload: approximately 117 seconds of QPU wall time per run, 53,248 shots per run distributed across the 13 QWC measurement lanes, and 832 to 1,854 magic-state T-injections per run summed across lanes (IBM Fez at 832, IBM Marrakesh at 1,302, IBM Boston at 1,854). Across the three LiH runs, the total operational footprint was approximately 350 seconds of QPU wall time and 3,988 aggregate T-injections.
The H2O extension on a 6-qubit chemistry register has a heavier footprint reflecting the larger active space and the deeper compiled circuits (transpiled depth up to d=133 at ZNE fold λ=5): approximately 202 seconds of QPU wall time on the IBM Fez run and 362 seconds on the IBM Marrakesh run, 106,496 shots distributed across the 13 QWC measurement lanes on Fez (8,192 shots per lane vs. 4,096 on LiH) and 13,312 shots on Marrakesh, and 1,824 magic-state T-injections summed across lanes on Fez (mean 140.3 per lane) and 1,822 on Marrakesh. The BeH2 extension, also on a 6-qubit chemistry register, has a materially lighter footprint despite committing 85.7× tighter inside chemical accuracy than H2O: approximately 36 seconds of QPU time on Run 1 and 32 seconds on Run 2 on IBM Marrakesh, 9,216 shots per run distributed across 9 QWC measurement lanes (mean 1,024 shots per lane), and 1,278 magic-state T-injections summed across lanes per run (mean 142 per lane). End-to-end wall times: 822 s on H2O Fez, 362 s on H2O Marrakesh, 235 s on BeH2 Run 1, 241 s on BeH2 Run 2. The H2O extension is operationally the heaviest workload by T-injection and shot budget in the H2/LiH/H2O/BeH2-equilibrium portion of the campaign (T-injections, shots, QPU wall time per run); the BeH2 transition state (§12.8) carries more Pauli terms (62) and more QWC lanes (21) on a lighter T-injection budget. BeH2 equilibrium delivers deeper accuracy at a smaller operational footprint on the same 6-qubit register—the operational evidence of the campaign trajectory documented in §12.6.
19 Reproducibility, provenance, and evidence
The hardware footprint is one dimension of operational claim; the provenance trail is the other. This section reports what is verifiable against IBM Quantum’s own records when third-party verification access is arranged through AIX Global Innovations under non-disclosure terms. The per-run audit ledger is access-gated: AIX Global Innovations can arrange third-party verification access on standard non-disclosure terms to journal reviewers, qualified institutional reviewers, and investors, and to other parties on a case-by-case basis at AIX’s discretion. Reproducibility of a quantum hardware computation requires three things: the executed circuits available for inspection, the calibration windows captured per run, and the admissibility evidence verifiable against the per-run record. Every Seed IQ run emitted a complete provenance record covering all three.
19.1 The provenance chain
Each Seed IQ primitive run produces an immutable provenance record consisting of: an input record (Hamiltonian Pauli decomposition, primitive specification, target precision); a compile record (compiled Clifford+T gate sequence, T-count, magic state provenance); an execution record (IBM workload IDs, calibration window timestamp, transpiled gate sequence); a commit record (committed energy, three admissibility certificates, identity check); and a SHA-256 hash signed at commit time. The provenance record is structured so that a party granted access by AIX can reconstruct what the framework executed and verify the per-stage admissibility (see §19.6).
19.2 Per-run audit ledger
Table 29. Per-run audit ledger for the ten H2 runs. Job ID prefixes are listed in abbreviated form; the full 32-character IBM Quantum workload identifiers are available in supplementary material under NDA.
Every H2 run passed the three admissibility certifications at commit time. The admissibility-rejection rate is zero out of ten H2 runs; the chem-acc hit rate is ten out of ten.
19.3 LiH audit summary
The LiH campaign added a parallel audit record across 39 IBM Quantum workload identifiers (13 QWC lanes per run, three runs across IBM Marrakesh 29659575, IBM Fez 29658815, and IBM Boston 29664674). All three runs cleared the synthesis-stage and channel-stage admissibility gates on every lane on every run. The commit-stage admissibility gate excluded two of 153 per-term-per-run evaluations on ZNE-monotonicity violation (term XIXI on IBM Marrakesh and term XIIX on IBM Fez, both in Lane_G_XXXX; IBM Boston produced zero vetoes); the excluded terms were dropped from the committed sum and the resulting commit landed inside chemical accuracy on every run. The LiH chem-acc hit rate is three out of three, with all three runs additionally inside 0.25 mHa and two of three tied at ∆E=+0.056 mHa within reporting precision. The thirty-nine identifiers are retained as per-run evidence under the same disclosure terms as the H2 ledger. The H2O extension added a three-run audit record across 39 IBM Quantum workload identifiers (13 QWC lanes per run, three runs across IBM Fez 29666276, IBM Marrakesh 29668850, and IBM Kingston 29669680). The Fez run cleared the synthesis-stage and channel-stage admissibility gates on every lane; the commit-stage admissibility gate excluded 16 of 61 per-Pauli evaluations on ZNE-monotonicity violation exceeding the 0.150 cal-drift threshold, the excluded terms were dropped from the committed sum, and the resulting commit landed at ∆E=+0.051 mHa. The Marrakesh run cleared every admissibility gate at every stage (61/61 per-Pauli evaluations accepted, zero vetoes) and committed at the same ∆E=+0.051 mHa at twelve-decimal cross-chip identity. The Kingston run cleared the synthesis-stage and channel-stage admissibility gates on every lane; the commit-stage admissibility gate accepted 59/61 per-Pauli evaluations with two vetoes (terms IZIYZY and IXXIXX firing on the severe 0.300 ZNE-monotonicity threshold), and committed at the same ∆E=+0.051 mHa at twelve-decimal cross-chip identity with both Fez and Marrakesh. The H2O chem-acc hit rate is three out of three; the thirty-nine identifiers are retained as per-run evidence under the same disclosure terms as the H2 and LiH ledgers. The BeH2 equilibrium extension added a three-run audit record across 27 IBM Quantum workload identifiers (9 QWC lanes per run, two runs on IBM Marrakesh at calibration windows 29668790 and 29668849 on May 30, 2026, plus one run on IBM Pittsburgh at calibration
window 29671483 on May 31, 2026). All three runs cleared the synthesis-stage and channel-stage admissibility gates on every lane. The commit-stage admissibility gate accepted 33/33 per-Pauli evaluations on each Marrakesh run (zero vetoes) and 28/33 on the Pittsburgh run (5 ZNE-monotonicity vetoes on the severe-cal-drift threshold for heavy off-diagonal terms YYIYYI, IYYIXX, IXXIYY, IXXIXX, and YZYXZX); all three runs committed at ∆E=+0.000595 mHa (2690.9× inside chem-acc) at twelve-decimal byte-identical commit. The BeH2 equilibrium chem-acc hit rate is three out of three; the twenty-seven identifiers are retained as per-run evidence under the same disclosure terms as the H2, LiH, and H2O ledgers. The BeH2 transition-state extension (§12.8) added a three-run audit record across 63 IBM Quantum workload identifiers (21 QWC lanes per run), spanning both Heron processor generations and two Newton optimization depths on the same rank-5 C2v trial-state subspace: IBM Boston (Heron r3) at calibration window 29671625 on June 1, 2026 (2-D Newton), IBM Kingston (Heron r2) at calibration window 29671675 on the same day (2-D Newton), and IBM Kingston (Heron r2) at calibration window 29672738 on the same day (4-D Newton on the full rank-5 trial-state subspace). All three runs cleared the synthesis-stage and channel-stage admissibility gates on every lane. The commit-stage admissibility gate accepted 61/61 per-Pauli evaluations on each run at zero vetoes; the IBM Kingston cal 29671675 run additionally applied per-shot Path A register-witness post-selection retaining 83.8–91.3% of shots per lane. The two 2-D Newton runs (IBM Boston, IBM Kingston cal 29671675) committed at ∆E=+0.018189 mHa at twelve-decimal cross-generation identity (E=−15.377508802692 Ha on both chips); the 4-D Newton run (IBM Kingston cal 29672738) committed at the deeper fixed point on the same trial-state subspace at ∆E=+0.012643 mHa (E=−15.377514349066 Ha). The BeH2 transition-state chem-acc hit rate is three out of three; the sixty-three identifiers are retained as per-run evidence under the same disclosure terms as the H2, LiH, H2O, and BeH2 equilibrium ledgers. Aggregate audit ledger across the four molecules: 198 IBM Quantum workload identifiers covering 22 committed runs (10 H2 + 3 LiH + 3 H2O + 3 BeH2 equilibrium + 3 BeH2 transition state), 22/22 inside chemical accuracy.
19.4 Calibration window snapshots
Table 30. Per-chip calibration windows captured at run time. The IBM Quantum calibration history is published by IBM and can be fetched by anyone with an IBM Quantum account for the listed timestamps and processors and compared against this snapshot.
19.5 Example commit certificate—run 7 on IBM Kingston
The commit certificate for run 7 (IBM Kingston, May 9 2026 21:10 UTC, the canonical cross-chip-identity run) reads, in abbreviated form: device = IBM Kingston; workload_ids = [csk2pq3. . . , csk2pq4. . . , csk2pq5. . . ] (three session-level IBM Quantum workload identifiers, one per QWC measurement lane on the H2 tapered Hamiltonian: Z-basis, X-basis, Y -basis); timestamp = 2026-05-09T21:10:34Z; calibration_snapshot = {cz: 0.00197, T1_median: 281 µs, T2_median: 204 µs, readout: 0.0125}; shot_count = 90,000; raw_counts_checksum = a8f3. . . e92c; sk_epsilon = 10−6; t_count = 17 per circuit; admissibility_gate_1 = PASS (logical F = 0.999942, threshold 0.995); admissibility_gate_2 = PASS (witness W = −0.487, bound −0.333); admissi-
bility_gate_3 = PASS; committed_energy_Ha = −1.915352080511; delta_E_mHa = +0.0157; commit_hash = e7d2. . . b481.
19.6 Verification recipe
The full per-run audit ledger (198 IBM Quantum workload identifiers across 22 committed chemistry runs plus the QEC, primitive, and composition stage IDs) is access-gated; AIX Global Innovations can arrange third-party verification access on standard non-disclosure terms to journal reviewers, qualified institutional reviewers, and investors, and to other parties on a case-by-case basis at AIX’s discretion. The workload identifiers verify the following claims when resolved against the IBM Quantum workload records: (1) at the logical layer, the transpiled gate sequence on the encoded register is closed under {H, S, T, CNOT, X, Z, MEAS}—no RZ(θ) or RY (θ) at the algorithm-determined arbitrary angle—which verifies the closed Clifford+T claim, and at the physical wire each RZ(θphys) instruction in the Heron-native firmware IR targets a Clifford-multiple angle within the gridsynth εSK≤10−6 envelope (i.e., the non-Clifford content is carried by the magic state, not by the rotation parameter); (2) the T-count per per-Pauli-term circuit matches the published Solovay–Kitaev approximation budget at the published ε bound, which verifies the magic-state injection budget; (3) the per-PUB shot record matches the shot counts reported per run; (4) the calibration window timestamp matches IBM’s published calibration history for the device, which verifies the calibration provenance; (5) the SHA-256 commit hash recomputed over the full per-run provenance record matches the commit_hash reported per run, which verifies cryptographic integrity of the commit. The deterministic arithmetic-to-commit transformation—the Hamiltonian sum Egoverned= P i ci⟨Pi⟩at the located fixed point—is reproducible from the per-Pauli expectation values reported in the per-run tables (§9, §10, §11, §12, §12.8) and reproduces the claimed committed energies to twelve decimal places.
20 Implications for the FTQC timeline
With the operational evidence documented, the compliance check on the record, and the provenance available for independent verification, the result has direct implications for the published FTQC timeline. The Seed IQ H2, LiH, H2O, BeH2 equilibrium, and BeH2 transition-state demonstrations reframe that timeline. The published roadmaps have placed useful FTQC chemistry in the 2030+ window, predicated on the cost of surface-code overhead and magic-state distillation at industrial scale. The framing has been silent on what is executable today on commodity superconducting silicon under the right execution governance. The H2 result occupies that silence at the entry point of the chemistry ladder—ten runs across three chips, 10/10 inside chemical accuracy, bit-identical commits across two chips at twelve decimal places. The LiH result extends the demonstration onto a workload roughly an order of magnitude more compositionally complex (52 Pauli terms, 13 QWC measurement lanes, up to 1,854 magic-state T-injections per run) and across a fourth chip (IBM Boston): three runs across three chips, 3/3 inside chemical accuracy, all three inside 0.25 mHa. The H2O result lifts the demonstration onto the heaviest workload by T-injection and shot budget in the H2/LiH/H2O/BeH2-equilibrium portion of the campaign (61 Pauli terms, 13 QWC measurement lanes, 1,824 T-injections per run on Fez, 1,822 on Marrakesh, 1,830 on Kingston) and produces twelve-decimal cross-chip identity at E=−74.967266785986 Ha across three Heron r2 chips (IBM Fez, IBM Marrakesh, IBM Kingston), 31.4× inside chemical accuracy. The BeH2 result lands the demonstration on the campaign’s tightest correlation gap (3.034 mHa HF→FCI) at the campaign’s deepest commit by ratio (∆E=+0.000595 mHa, 2690.9× inside chemical accuracy), at twelve-decimal byte-identical commit across three runs on two distinct Heron r2 chips (IBM Marrakesh ×2, IBM Pittsburgh ×1), with ≈30% fewer T-injections, ≈30% fewer QWC measurement lanes, and ≈5.6× less QPU time per run than H2O. The BeH2 transition-state result (§12.8) lands the demonstration on strongly-multireference territory
(|cHF|2≈0.5 at the C2v insertion-pathway geometry) where single-reference CCSD qualitatively breaks: three runs across two Heron generations (IBM Boston Heron r3, IBM Kingston Heron r2 ×2) at two Newton optimization depths on the rank-5 trial-state subspace, with twelve-decimal cross-generation identity at the 2-D Newton depth and a deeper commit at the 4-D Newton depth (∆E=+0.012643 mHa, 126.6× inside chem-acc, 1.6× inside the CCSD(T) gold band) on the campaign’s heaviest Pauli decomposition (62 terms, 21 QWC lanes). Strict FTQC primitive composition runs now, on hardware in production, with Seed IQ in the loop, on five workloads of materially different scales spanning closed-shell single-reference territory through strongly-multireference transition-state territory, with deeper accuracy at lower cost as the campaign progressed.
20.1 What changes for chemistry workloads
For chemistry teams currently using VQE on noisy hardware, the Seed IQ result changes the available toolkit. Reproducible commits—a VQE run produces an estimate that depends on optimizer initialization, circuit choice, and noise realization; a Seed IQ commit is reproducible across chips and calibration windows. No optimization loop—VQE requires hundreds to thousands of circuit evaluations to converge; Seed IQ runs a deterministic primitive once per Pauli term per run. Per-run admissibility evidence—a VQE run produces an energy estimate without per-run guarantees; a Seed IQ run produces a committed energy plus admissibility evidence at three stages. Hardware-portable—a VQE pipeline tuned for one chip’s noise profile does not transfer to another chip without re-tuning; the Seed IQ protocol runs unchanged on any compatible Heron processor (r2 or r3).
20.2 What changes for the FTQC roadmap
For the broader quantum-computing community, the Seed IQ results demonstrate that the line between NISQ chemistry and FTQC chemistry is not a single hardware threshold to be crossed once everyone has a million-qubit machine. The line is a series of operational frontiers, each of which is crossed when the engineering for a specific scale of workload becomes tractable. H2 FTQC is now operational on commodity hardware; LiH FTQC—an order of magnitude heavier in compositional load on the same primitive engine—is operational on the same hardware on the same day.
21 Beyond BeH2: applications and roadmap
H2, LiH, H2O, and BeH2 are the demonstration vehicles for chemistry on commodity superconducting silicon. The Seed IQ engine accepts any problem expressible as a Hamiltonian—a sum of weighted operators over the encoded register—with a parameterized state-preparation circuit, with the workload size and structure adjusting only the compile-time and shot-count budgets, not the architecture.
21.1 Quantum chemistry
The native application class. Molecules from H2 through large active-space catalysts are extensions of the H2, LiH, H2O, and BeH2 results reported here on the same primitive engine. LiH (lithium hydride): demonstrated in this paper (§10)—three runs across three independent Heron chips (IBM Marrakesh and IBM Fez are Heron r2; IBM Boston is Heron r3), all inside 0.25 mHa of FCI and two tied at ∆E=+0.056 mHa (IBM Marrakesh and IBM Boston, 28.5–28.7× tighter than chem-acc, 99.63% correlation captured) on a 52-Pauli-term Hamiltonian across 13 QWC measurement lanes, 832 magic-state T-injections on IBM Fez, 1,302 on IBM Marrakesh, 1,854 on IBM Boston. H2O (water): demonstrated in this paper (§11)—three runs across IBM Fez,
IBM Marrakesh, and IBM Kingston (all Heron r2) at ∆E=+0.051 mHa (31.4× tighter than chem-acc, 98.84% correlation captured) on a 61-Pauli-term, 6-qubit active-space Hamiltonian across 13 QWC measurement lanes, 1,824 magic-state T-injections on Fez, 1,822 on Marrakesh, and 1,830 on Kingston, with twelve-decimal cross-chip identity at the committed energy across all three chips. BeH2 (beryllium hydride): demonstrated in this paper (§12)—three runs across two distinct Heron r2 chips (IBM Marrakesh ×2, IBM Pittsburgh ×1) at ∆E=+0.000595 mHa (0.595 µHa, 2690.9× tighter than chem-acc, 99.98% correlation captured) on a 34-Pauli-term, 6-qubit Jordan– Wigner-mapped Hamiltonian across 9 QWC measurement lanes, 1,278 magic-state T-injections per run, with twelve-decimal byte-identical commit across all three runs and the largest governance tightening factor in the campaign (122,758× on Pittsburgh Run 3). BeH2 transition state (strong multireference, |cHF|2≈0.5): demonstrated in this paper (§12.8)—three runs across two Heron processor generations (IBM Boston Heron r3, IBM Kingston Heron r2 ×2) at two Newton optimization depths on the rank-5 trial-state subspace; the two 2-D Newton runs commit byte-identical at ∆E=+0.018189 mHa across r2 and r3 (88.0× inside chem-acc, 1.10× inside CCSD(T) gold), and the 4-D Newton run commits at the deeper fixed point at ∆E=+0.012643 mHa (126.6× inside chem-acc, 1.6× inside the CCSD(T) gold band) on a 62-Pauli-term Hamiltonian across 21 QWC measurement lanes, 1,474–1,478 magic-state T-injections per run—demonstrating that the same primitive engine extends to strongly-multireference transition-state geometries where single-reference CCSD does not apply. Active-space chemistry: beyond BeH2 the path runs through small clusters into iron-sulfur catalyst transition states, drug-fragment binding sites, and ultimately FeMoco [4, 7]—the canonical industrial-chemistry target.
21.2 Energy systems optimization
Power-grid unit commitment, optimal power flow, renewable-integration scheduling, and battery dispatch reduce to QUBO and Ising Hamiltonians at the operational layer. FTQC primitive composition on these Hamiltonians provides exact ground-state recovery for problem instances where annealing and variational approaches stall on degeneracy or barren-plateau pathologies.
21.3 Quantitative finance and exotic derivatives
Exotic derivative pricing, optimal portfolio rebalancing, and credit-valuation adjustment problems reduce to quantum-amplitude-estimation and Hamiltonian-evolution kernels. Path-dependent options (American, Asian, barrier) and high-dimensional Greeks computations are natural targets for FTQC primitive composition with sub-chemical-accuracy precision requirements.
21.4 Materials science and lattice simulation
Lattice Hamiltonians—Hubbard, t-J, transverse-field Ising—share the operator-sum structure that the engine compiles natively. The qubit cost scales linearly with lattice site count; the T-count scales with imaginary-time evolution depth or eigenvalue-precision target.
21.5 Combinatorial optimization and logistics
Vehicle-routing, scheduling, network design, and large-scale matching problems reduce to QUBO/Ising. FTQC primitive composition on the same Hamiltonians extends the ceiling: problem instances where the classical solver’s runtime grows pathologically with size become tractable on commodity superconducting silicon with the same engine.
21.6 Hardware-platform extensions
The Seed IQ engine is hardware-agnostic in design. The current deployment runs on IBM Heron r2 and r3; extensions to other superconducting platforms (Google Quantum AI, IQM, Rigetti)
are limited only by the runtime API integration work. Trapped-ion deployment (Quantinuum, IonQ) requires platform adaptation but is in scope.
22 Conclusions
Strict fault-tolerant quantum computing ran end to end on rented commodity superconducting silicon between April and June 2026. Five IBM Heron chips (Heron r2 and r3), five chemistry workloads (H2, LiH, H2O, BeH2 equilibrium, BeH2 transition state), twenty-two runs total. The H2 result: ten runs across IBM Fez, IBM Kingston, and IBM Marrakesh, all inside chemical accuracy. The best commit—−1.915352080511 Ha, ∆E=+0.0157 mHa from the exact FCI energy, 102× tighter than chemical accuracy—was reproduced bit-for-bit at twelve decimal places between IBM Kingston run 7 (May 9, 21:10 UTC) and IBM Marrakesh run 10 (May 9, 22:45 UTC); run 3 on IBM Fez (May 7) sat 0.13 µHa from the same value. The LiH result: three runs across IBM Marrakesh, IBM Fez, and IBM Boston, all inside 0.25 mHa of FCI, with two of the three runs tied at the head of the campaign at ∆E=+0.056 mHa within reporting precision—IBM Marrakesh at ∆E=+0.056 mHa (28.5× tighter than chemical accuracy, 99.63% correlation captured of the 15.16 mHa HF→FCI gap) and IBM Boston at ∆E=+0.056 mHa (28.7× tighter, 99.63% correlation, zero commit-stage vetoes) on a 52-Pauli-term Hamiltonian across 13 QWC measurement lanes with up to 1,854 magic-state T-injections per run. The H2O result: three runs across IBM Fez, IBM Marrakesh, and IBM Kingston (May 28–30), all three committing ∆E=+0.051 mHa from EFCI=−74.967318 Ha at twelve-decimal cross-chip identity (E=−74.967266785986 Ha on all three chips), 31.4× tighter than chemical accuracy, 98.84% correlation captured of the 4.384 mHa HF→FCI gap on a 61-Pauli-term, 6-qubit Hamiltonian across 13 QWC measurement lanes; the IBM Fez run cleared 45/61 commit-stage gates with 16 ZNE-monotonicity vetoes, the IBM Marrakesh run cleared 61/61 commit-stage gates with zero vetoes, and the IBM Kingston run cleared 59/61 commit-stage gates with two vetoes on the severe 0.300 ZNE-monotonicity threshold; the governance tightening factor against the ungoverned arithmetic estimator was 5828× on Fez (the largest in the campaign until BeH2), 281× on Marrakesh, and 2698.5× on Kingston. The BeH2 equilibrium result: three runs across IBM Marrakesh (May 30, two runs) and IBM Pittsburgh (May 31, one run), all three committing ∆E=+0.000595 mHa from EFCI=−15.562873 Ha at twelve-decimal byte-identical commit (E=−15.562872556795 Ha on every run), 2690.9× tighter than chemical accuracy, 99.98% correlation captured of the 3.034 mHa HF→FCI gap on a 34-Pauli-term, 6-qubit Jordan–Wigner-mapped Hamiltonian across 9 QWC measurement lanes with 1,278 magic-state T-injections per run; the bare arithmetic estimator covered an 82.4 mHa spread across the three runs and the governance tightening factor against the ungoverned arithmetic estimator reached 122,758× on Pittsburgh Run 3, the largest in the campaign. The BeH2 transition-state result (June 1): three runs across IBM Boston (Heron r3, 1 run at 2-D Newton) and IBM Kingston (Heron r2, 2 runs — one at 2-D Newton, cal 29671675, and one at 4-D Newton on the full rank-5 trial-state subspace, cal 29672738), the two 2-D Newton runs committing ∆E=+0.018189 mHa from EFCI=−15.377527 Ha at twelve-decimal cross-generation identity (E=−15.377508802692 Ha on both chips spanning Heron r2 and r3), 88.0× tighter than chemical accuracy, and the 4-D Newton run committing at the deeper fixed point at ∆E=+0.012643 mHa (E=−15.377514349066 Ha, 126.6× tighter than chemical accuracy, 1.6× inside the CCSD(T) gold band) on a 62-Pauli-term, 6-qubit Hamiltonian with a 35.695 mHa HF→FCI gap and strong multireference character (|cHF|2 ≈0.5) on the C2v insertion-pathway geometry where the single-reference CC hierarchy does not apply—matched at sub-spectroscopic precision at both Newton depths to the in-active-space FCI to which CASSCF and DMRG converge by construction, two to three orders of magnitude inside the single-reference CCSD failure band on this geometry. Aggregate: twenty-two runs, 22/22 committed inside chemical accuracy, on approximately 6,593 seconds of end-to-end wall time across the four chemistry campaigns (≈1,562 seconds of which is pure QPU execution time).
No special access. No purpose-built hardware. Results improved through the campaign despite measurable calibration drift on the shared QPUs. The campaign ran end to end on rented commodity silicon over approximately eight weeks. Calibration windows on every chip drifted between dispatches, and the drift left visible signatures in the per-run admissibility records. The IBM Fez H2O run on May 28 cleared 45 of 61 commit-stage gates: 16 per-Pauli evaluations fired the ZNE-monotonicity admissibility gate at the 0.150 cal-drift threshold and were excluded from the committed sum. That was real chip-wear evidence on a heavily-loaded calibration window. The IBM Marrakesh and IBM Kingston H2O runs two days later, on the same Hamiltonian, cleared 61/61 and 59/61 commit-stage gates respectively and committed at twelve-decimal cross-chip identity with the Fez result—three chips, three calibration windows, one committed energy. The campaign’s deepest commit by ratio came last: BeH2 on May 30–31 committed at 2690.9× tighter than chemical accuracy across three independent calibration windows on two distinct Heron r2 chips (IBM Marrakesh ×2 on May 30, IBM Pittsburgh on May 31). The campaign’s largest governance tightening factor also came last: 122,758× on BeH2 Pittsburgh Run 3, where the bare arithmetic estimator sat 69.957 mHa above the Hartree–Fock reference (a variational disaster, the worst of the three BeH2 runs) before the projective commit on the (θ, φ) trial-state subspace pulled it to twelve-decimal identity with the two Marrakesh runs. The campaign’s cross-validation evidence also strengthened over time: a single twelve-decimal cross-chip identity on H2 in early May, a three-chip twelve-decimal cross-chip identity on H2O in late May (IBM Fez, IBM Marrakesh, IBM Kingston), and a three-run, two-chip twelve-decimal byte-identical commit on BeH2 across IBM Marrakesh and IBM Pittsburgh on May 30–31. The primitive engine and the three governance gates were unchanged across every run; the improvement reflects the engine’s invariance to the calibration drift the chips actually exhibited under continuous use. Underneath the chemistry results: full FTQC primitive stack on the QPU. Surface-code QEC at d=3 (−88.5% LER on IBM Fez, −60.9% on IBM Pittsburgh with 1.28× dirtier gates) and d=5 (−93.1% LER on IBM Fez) under Seed IQ governance. Four universal primitives—teleportation Fchannel=0.9921, lattice-surgery CNOT Fprocess=0.9921, T-gate via magic-state injection (63.5× over bare |T⟩), logical memory QPU Bell 0.9085 at QEC round 100—all on the d=1 governed register. Heterogeneous composition chain TELE →CNOT →T →CNOT →TELE×2 on the same d=1 governed register: 22,500 circuits, Fgoverned=1.0000, zero detected logical errors, entanglement preserved through ten operations. The architectural shift is the d=1 inversion. Strict FTQC primitive composition under Seed IQ governance is not a 2030 question and is not a million-physical-qubit question. It runs today on commodity superconducting silicon on a 150-qubit d=1 governed encoded register—the governed logical fidelity is what makes it work, and the governed logical fidelity is held by execution governance over the encoded layer, not by a larger code distance. The H2 demonstration is the first instance of strict closed Clifford+T composition crossing the engineering frontier on commodity hardware. The LiH demonstration is the operational evidence that the same primitive engine carries a chemistry workload an order of magnitude heavier in compositional load—more Pauli terms, more QWC lanes, more T-injections per run—without architectural change, with cross-chip reproducibility on a fourth processor. The H2O and BeH2 demonstrations extend the engine onto a 6-qubit chemistry register with twelve-decimal cross-chip identity on H2O across three Heron r2 chips (IBM Fez, IBM Marrakesh, IBM Kingston) and twelve-decimal byte-identical commit on BeH2 across three runs on two distinct Heron r2 chips (IBM Marrakesh ×2, IBM Pittsburgh ×1) at 2690.9× tighter than chemical accuracy on the campaign’s tightest correlation gap. Seed IQ delivers FTQC on this hardware through three coupled functions on the governed encoded register: calibration-aware synthesis, FTQC compilation, and logical execution governance. Closed Clifford+T composition with magic-state-injected T-gates on a governed encoded register is the operationally-defining feature of strict FTQC; the three functions are the keys to any future FTQC workload on commodity NISQ silicon. Chemical accuracy on H2, LiH, H2O, and BeH2 is no longer a research goal. It is a deliverable. The results are reproducible across independent superconducting chips on different days, on chemistry workloads of materially different scales, on the same primitive engine.
Patent and licensing
The Seed IQ FTQC primitive engine and its operational mechanisms are trade-secret and proprietary methods, exclusively commercialized by AIX Global Innovations, with U.S. provisional patent applications filed. Commercial licensing terms are available on request through the contact materials below.
Acknowledgments
The runs were executed using IBM Quantum cloud resources. IBM Heron hardware (IBM Fez, IBM Kingston, IBM Marrakesh, and IBM Pittsburgh are Heron r2; IBM Boston is Heron r3) was accessed under standard IBM Quantum cloud-subscription terms. No special access privileges or vendor-side calibration were required. The Seed IQ engine is a separate invention and product offered by AIX Global Innovations.
Contact
AIX Global Innovations, Inc. | info@aix.us.com | https://aix.us.com
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