From Commutants to Logical Control: Pre-Registered Verification of a Decoherence-Free Subspace on IBM Heron Processors

Protected Quantum Information on IBM Heron Processors: A Four-Experiment Pre-Registered Arc — Commutant Verification, Noise-Axis Characterization, Subspace Certification, and Logical Operation Under Interleaved Collective Rotations Date: July 10, 2026 Version: 2.1 (incorporates independent post-hoc review corrections; supersedes drafts 1.0/2.0) Related records: Phase 1 spatial Y⊗Z parity-consistency (DOI: 10.5281/zenodo.18501679); correction record v1.3 (DOI: 10.5281/zenodo.20470129) Plain-language summary Quantum computers are extremely sensitive to noise — stray influences that corrupt the information they hold. But not all quantum information is equally fragile. If you store information in the right pattern across two qubits, certain kinds of disturbance pass through without changing it, the way a perfectly symmetric object looks unchanged in a mirror. Over four experiments on two IBM quantum processors (roughly two and a half minutes of total quantum computer time, as reported by our submission records), we asked four questions in order: Are the sign changes predicted by the protection math reproduced on real hardware? Yes. In all 42 individual circuit-level tests, deliberately applied disturbances flipped the stored signal exactly when the math said they would, and left its sign unchanged when the math said they should, with high measured contrast. Does the machine's own natural noise happen to be the kind this protection guards against? On the qubits we tested, no — and that is an important honest finding. The measured decay was consistent with energy leakage dominating over phase scrambling, so this particular protection does not help against the everyday noise we observed. We predicted the numbers in advance, and where our model was wrong, the record shows exactly what we corrected and when. Can a protected "safe pattern" be demonstrated directly? Yes. We prepared two states of similar complexity — one inside the protected pattern, one outside — and applied the same programmed disturbance to both. Across the whole disturbance range, the protected state showed no statistically resolved response under our registered test, while the unprotected one was driven all the way to complete inversion, exactly as predicted. Can you operate on information inside the safe pattern while disturbances are applied between the steps? Yes. We ran a small logical operation sequence — preparing information, rotating it, flipping it, and reading out the answer — with programmed disturbances interleaved between logical stages. The protected version's answers showed no systematic dependence on the disturbances; the same rotation-and-readout schedule applied to the unprotected pattern returned answers corrupted by exactly the amount the theory predicted, including cases where it gave the opposite of the truth. One caution about what "disturbance" means here: our injected disturbances were deterministic, programmed rotations chosen to mimic the symmetry of a real noise process — not random noise itself. That is the right tool for testing the protection's symmetry, and we say so plainly. None of this is new physics — the theory dates to 1997 and was first demonstrated in other hardware in the early 2000s. What this record contributes is the method: every numerical prediction was written into a manifest before the hardware job was submitted, every circuit was committed by SHA-256 hash for later integrity verification, and every design mistake and model error we made along the way (there were several) is documented rather than hidden. The result is a small but complete example of how quantum hardware claims can be made verifiable — which is the business Quantum Clarity is in. 1. Overview and claims Four pre-registered experiments, executed July 9–10, 2026: # Job ID Backend Qubits Question Outcome 1 d9897st2su3c739j056g ibm_marrakesh [35, 6] Are commutation-predicted signs reproduced on hardware? Yes: 36/36 circuit-level deterministic sign checks agreed, min |parity| ≈ 0.961 2 d98h1ncqp3as739sus3g ibm_marrakesh [35, 6] Is native idle decoherence on the selected qubits Z-biased? Echoed decays consistent with relaxation-dominated decoherence; Exp.-1 confound measured as 14.27 ± 0.45 kHz static detuning 3 d98mnod2su3c739jg280 ibm_fez [2, 3] Is span{|01⟩,|10⟩} protected against collective Z rotations, within measurement precision? Yes: protected state showed no resolved collective-angle dependence (max 2.15σ) while the mirror state traced cos 2θ through full inversion 4 d98n5hcqp3as739t60fg ibm_fez [2, 3] Do logical operations inside the DFS show dependence on interleaved collective rotations? No systematic dependence detected (registered max-spread ≤ 2.42σ); mirror-sector control followed the predicted phase response Central scoped claim (wording follows independent post-hoc review): On the tested IBM Heron-class qubits, the exact decoherence-free-subspace model for span{|01⟩,|10⟩} was confirmed within finite-shot measurement precision under deterministic collective-Z rotations. Protected-state observables and a shallow logical Ramsey operation showed no systematic dependence on the registered collective rotations, while a mirror-sector control followed the predicted phase response. The encoding showed no detectable lifetime advantage under the tested native echoed-decay conditions; its demonstrated protection is conditional on the collective-Z symmetry and does not extend to arbitrary or predominantly local hardware noise. 2. Theoretical basis (established physics) An observable O is unaffected by a channel with Kraus operators {Kᵢ} iff [O, Kᵢ] = 0 for all i (Zanardi & Rasetti 1997; Knill, Laflamme & Viola 2000; Lidar et al.). For |0_L⟩ = |01⟩, |1_L⟩ = |10⟩: a collective RZ(θ)⊗RZ(θ) applies equal and opposite phases to the code words and acts as the identity on the code space, while acting on the mirror sector {|00⟩,|11⟩} as a logical rotation of angle 2θ. Logical operators: Z_L = Z⊗I (a differential single-qubit RZ is an exact logical RZ_L in the ideal model), X_L = X⊗X. First hardware demonstrations of DFS logic: Kwiat et al. 2000 (photons); Kielpinski et al. 2001–2002 (trapped ions). This record's contribution is methodological, not physical novelty. 3. Methods common to all experiments Qiskit Runtime SamplerV2. Product-state or single-Bell-pair preparations only; Experiments 1–2 contain no two-qubit gates; Experiments 3–4 contain one CZ-equivalent per circuit. All injected "disturbances" are deterministic virtual RZ frame updates with no scheduled pulse duration — programmed rotations sharing the symmetry of collective dephasing, not stochastic noise. Dual provenance for every circuit: full QASM3 stored privately; SHA-256 hash deposited in a manifest that also contains all numerical predictions, written before job submission (manifest and submission timestamps are recorded in the files; no externally anchored cryptographic timestamp is claimed). From Experiment 2 onward, each circuit's ideal expectation was checked against a statevector computation before submission (author-reported; the check code is withheld, see §7). Runtime options (dynamical decoupling and gate twirling disabled) are author-reported per the withheld submission code and recorded in manifest design notes. 4. Results All parities below reproduce from the deposited raw counts; an independent post-hoc review verified one-to-one manifest/result correspondence, job IDs, shot totals, and the recomputation of every reported statistic, and its corrections are incorporated throughout. 4.1 Experiment 1 — deterministic commutant sign verification Injected Z operations on the Y leg of Y⊗Z flipped the measured parity sign per insertion ((−1)^(n_Z)); injections on commuting legs left the sign unchanged. 36/36 circuit-level checks agreed (min |parity| ≈ 0.961 through 20 insertion rounds, no depth-dependent degradation), reproduced with deterministic insertion counts in Experiment 2: 6/6. Total 42/42. Status: positive control — Pauli algebra is not in question; the value is certification of the preparation/injection/rotation/ readout/analysis chain. A separately registered two-randomization ensemble-decay analysis in this job was invalid by design and is withdrawn (§5.1). 4.2 Experiment 2 — native noise character on the selected qubits CPMG-2 echoed decays (pulses only on legs anticommuting with Z; even pulse count preserves the measurement frame): Observable Fitted Calibration reference Y⊗Z T = 550.7 ± 22.8 µs T2(q35) = 461 µs Y⊗Y T = 156.6 ± 4.6 µs series-sum 171 µs Z⊗Z (prep |01⟩) T1 = 288.8 ± 7.3 µs T1(q6) = 358 µs The fitted Y-leg transverse rate was interpreted as having a smaller pure-dephasing contribution than relaxation contribution; the numerical decomposition depends on T1(q35), which was not recorded in the original manifest and is documented in the accompanying analysis supplement (analysis_supplement_zbias.json) rather than asserted here as a standalone reproducible ratio. A dense unechoed Ramsey series fit the static detuning at 14.27 ± 0.45 kHz (T2* = 231.4 µs), reproducing Experiment 1's unechoed idle-arm oscillation point-by-point and closing that confound quantitatively. The pre-registered ZZ model was falsified (wrong asymptote); the pre-observation amended model matched the observed functional form (relaxation toward +⟨Z⟩ with the observed zero crossing), though its calibration-based point values differed from the data at 5–12 standard errors because the fitted T1 was shorter than the calibration input — the amendment repaired the model's form, not its point predictions, and the record says so. 4.3 Experiment 3 — DFS certification under collective rotations Collective RZ(θ) sweep, θ ∈ {0…90°}, deterministic point predictions: Ψ⁺ (inside DFS): 0.964, 0.960, 0.952, 0.942, 0.951 across the sweep; no statistically resolved collective-angle dependence under the registered pointwise criterion (max 2.15σ vs baseline 0.9583). Φ⁺ (outside): 0.971, 0.694, 0.012, −0.629, −0.935 — tracing cos 2θ (fit amplitude 0.947 ± 0.013, phase offset −1.75°) through zero at 45° to full inversion at 90°. Differential falsifier: RZ(π) on one qubit drove the DFS state to −0.961 (mirror −0.958) as required — the protection is specifically collective. 2/2. Parity-sector checks: 4/4 retained expected signs within ≈3σ of the baseline model. Native-decay comparison (registered null): Ψ⁺ and Φ⁺ 240 µs echoed survivals ≈ 0.045 and 0.087; difference z = −1.28. No detectable DFS lifetime advantage at the tested delay — consistent with predominantly local noise and providing no evidence of a useful collective-Z component. (This comparison does not by itself uniquely establish noise independence or relaxation domination on this device; see §5.8.) 4.4 Experiment 4 — logical operation under interleaved rotations Logical Ramsey: prep |+_L⟩ (or the mirror |+_M⟩), collective kick C(φ₁), logical RZ_L(θ), collective kick C(φ₂), measure X_L. The same logical-rotation and collective-kick schedule was applied after preparing either the protected Ψ⁺ state or the mirror-sector Φ⁺ state (the preparations themselves differ by one X gate). Grid: θ ∈ {0°, 90°, 180°} × kick-sum ∈ {0°, 180°, 270°}. Protected register: no systematic kick dependence detected under the registered maximum-spread criterion; all protected-sector spreads ≤ 2.42σ (2.42σ, 2.22σ, 1.99σ at θ = 0°, 90°, 180°). A post-hoc weighted constant fit gives χ² = 6.54 (2 df, p ≈ 0.038) at θ = 0°, not significant across the three angle groups after multiplicity correction and with no kick-shaped structure. Mirror register: followed cos(θ + kick-sum) at all nine points (largest pointwise discrepancy: the disclosed 3.71σ contrast deficit, §5.6), including +0.936 where the uncorrupted answer is 0 and −0.939 where it is +1. Contrast pair (θ=0, kick=180°): protected +0.931 vs mirror −0.939, separation 1.87 under the same post-preparation schedule. Logical bit-flip control: X_L produced −0.992 vs the +0.993 identity reference (|⟨Z₀⟩| > 0.992 both). The two ZZ sector checks retained expected signs with |⟨ZZ⟩| ≥ 0.935; the mirror-sector check showed an additional contrast deficit relative to the baseline model (folded into §5.6). Echoed-decay model, registered prospectively: the model registered before this run, survival = e^(−2τ/166 µs) (effective 1/e scale 83 µs), was consistent with all eight points within 3σ (worst −2.27σ). A post-run weighted exponential fit to the same points gives ≈ 78.8 ± 2.4 µs (per state ≈ 80.6 / 77.1 µs), shot-noise uncertainties only. These are CPMG-2 echoed decay scales, not unrestricted raw-idle lifetimes. The Experiment-3 post-hoc model loop is thereby closed with a prospective confirmation of functional form and scale. 5. Deviations, errors, and amendments (complete ledger) Exp. 1 injected-arm design flaw. K = 2 stochastic randomizations cannot estimate ensemble decay; that analysis was invalid as designed and is withdrawn. The circuit-level deterministic sign test uses the same raw data and is unaffected. Exp. 1 missing echo. The unechoed idle arm conflated coherent detuning with dephasing; superseded by Exp. 2, which measured the confound (14.27 kHz). Exp. 2 ZZ model amendment (pre-observation, post-execution). Registered model used a wrong asymptote; the amended model was timestamped in the manifest before results were retrieved. The amendment repaired functional form only; its calibration-based point values still missed at 5–12 SE due to T1 drift/shortfall vs calibration. Both models' scores are archived. Exp. 2 self-test checker bug (self-caught). The pre-submission statevector check initially probed the wrong measurement frame; it flagged its own inconsistency on first execution and was corrected before any hardware use (author-reported; check code withheld). Exp. 3 native point-prediction miss (post-hoc model). The registered T1-limited prediction (0.325 at 240 µs) omitted pure dephasing; measured survivals were 0.045/0.087. The dephasing-sum model (0.056 at 240 µs) was post-hoc in Exp. 3, then registered prospectively in Exp. 4 and confirmed within 3σ. Exp. 3's registered hypothesis (equal decay of the two states) held regardless. Exp. 4 contrast deficit (noted, unmodeled). Several high-|parity| points sat 2–3% below the CZ+readout baseline model (worst 3.71σ; the v4 mirror ZZ check at ≈3.25σ is included here). Functional forms are unambiguous throughout; the deficit is left as a characterized residual. Bit-flip metadata omission. The v4 bit-flip circuits contain interleaved π/4 collective kicks in their (privately retained, hashed) QASM, but the kick angles were not registered in the public manifest metadata; the public record therefore supports only the bit-flip result itself, and the kicks are not claimed in §4.4's bit-flip statement. Scope limit on the noise-independence conclusion. Earlier drafts claimed native noise on both devices is "independent per qubit and relaxation-dominated." The deposited data support only: Marrakesh echoed decays consistent with relaxation-dominated decoherence on the selected qubits; and on fez, no detectable encoded-vs-mirror lifetime difference at the tested delay, consistent with predominantly local noise. Uniqueness of noise model is not established. Z-bias ratio provenance gap. The numerical decomposition (ratio ≈ 0.35) requires T1(q35), which was used in analysis but absent from the original manifest; it is documented in the analysis supplement with its source and formula rather than presented as independently reproducible from the original deposit. Edge-selection scoring corrections (pre-submission). Two qubit/edge-ranking heuristics mis-weighted figures of merit; both were caught and corrected before the affected submissions. Timestamp wording. Earlier drafts said predictions were "cryptographically time-stamped." SHA-256 hashes establish integrity, not time; the record claims only manifest-recorded creation and submission timestamps. 6. Conclusions Commutation-predicted sign behavior was reproduced on hardware in all 42 circuit-level checks with high measured contrast. On the selected Marrakesh qubits, echoed decay measurements were consistent with relaxation-dominated decoherence; on fez, encoded and mirror states showed no detectable native-lifetime difference at the tested delay. Neither Y⊗Z structure nor DFS membership showed advantage against the native decoherence observed. These nulls were pre-registered. Within finite-shot precision, the DFS span{|01⟩,|10⟩} protected both stored states and a shallow logical operation sequence against deterministic collective-Z rotations, while identical post-preparation schedules on the mirror sector were corrupted by precisely the predicted phase response. Not claimed: physical novelty; protection against native, stochastic, or arbitrary noise; uniqueness of any noise model; scalability beyond two qubits; externally anchored timestamps; and nothing herein uses or discloses proprietary basis-migration sequences. 7. Files in this deposit Manifests: yz_subspace_manifest.json, yz_subspace_v2_manifest.json (contains the pre-observation amendment), dfs_v3_manifest.json, dfs_v4_manifest.json. Result archives: results_<job_id>.json × 4. Figures: four results PNGs. Analysis supplement: analysis_supplement_zbias.json (T1(q35) input, source, and decomposition formula). Submission and analysis code is withheld pending patent counsel review; consequently, statevector self-test execution, disabled Runtime options, gate counts, and QPU-time figures are author-reported. The manifests' per-circuit SHA-256 hashes permit verification of any subsequently disclosed circuit or code against this record. Private QASM retained by the author. 8. Acknowledgments Quantum hardware access via IBM Quantum. Experiment design and analysis assisted by Anthropic's Claude; an independent post-hoc review of the deposited data (recomputation of all reported statistics) contributed the claim-precision corrections incorporated in this version. All experimental decisions, executions, and claims are the author's. License: CC-BY-4.0. Keywords: decoherence-free subspace, noiseless subsystem, logical qubit, collective dephasing, biased noise, pre-registration, falsification, provenance, Heron, null result.