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Sufficient quantum provenance: retained fields and certified recording precision

George Kourousias, Sergio Carrato, Roberto Pugliese, Aljoša Hafner, Francesco Guzzi, Matteo Billè, Fulvio Billè, Panagiotis Dimitrakis

arXiv:2610.09931v1quant-phcs.ET

TL;DR

Quantum reproducibility needs to specify which executions and outcome laws are being compared and at what tolerance. The paper certifies sufficient records using finite comparisons and a continuous channel bound, and reports a 13-bit QAOA-1 record along with different raw and decoded hardware label budgets.

  • Problem

    A reproducibility claim must specify which executions and outcomes it compares and what discrepancy is acceptable, because compilation and hardware conditions can change outcome laws without changing the logical task.

    Source: §I · p. 1

  • Method

    The framework defines record risk as the largest outcome-law distance hidden within a record cell and uses a channel-derived certificate to bound continuous-cell differences without simulating outcome distributions.

    Source: §III · p. 3, §VIII · p. 9

  • Results

    13 bits suffice for QAOA-1 with B < 0.049716, while fresh hardware comparisons require two labels for raw laws and one after a fixed decoder.

    Source: §III · p. 4, §VIII · p. 9

  • Takeaways & Limitations

    Comparison records specify tolerance and evidence class, but do not replace the detailed execution archive needed to investigate how a result was produced.

    Source: §VIII · p. 9

    Sufficient comparison records do not replace detailed execution archives, and the appropriate representation depends on whether the objective is comparison, attribution, or replay.

    Source: §VIII · p. 9

Abstract

from arXiv · show

Which execution details must a quantum computation retain, and to what precision, to support a declared comparison? We define a sufficient record by the largest Hellinger distance between outcome laws that share it. Our main result is a circuit-derived certificate for continuous recording precision: independent channel-mixture probabilities enter through the affinity of latent noise flags, while coherent rotation differences enter through conditional quantum fidelity. Their joint composition bounds whole record cells without simulating their outcome distributions. For four-qubit QAOA at depth one, a 13-bit comparison record guarantees distance below 0.05 throughout a declared continuous noise box, uniformly over programmed angles when comparisons hold the logical task fixed. A construction with observable noise flags attains the general bound. Exact tensorization of Hellinger affinity connects recording precision to the length of a future measurement transcript. On finite context classes, separation witnesses certify minimum-cost field retention; a four-field record is the unique minimum for 28 archived QAOA instances. Fresh six-qubit processor measurements establish different minimum label counts at one common tolerance: two labels are necessary and sufficient for raw laws, while one suffices after a fixed decoder for a synthetic image-segmentation task. These hardware conclusions are conditional on stationary independent shots and remain distinct from the analytic channel certificate. Shared task definitions and evidential archives are retained separately. The framework specifies which execution distinctions matter, how precisely to record them, and what evidence supports the resulting agreement guarantee.

I. FROM PRESERVATION TO A GUARANTEE

A reproducibility claim must say which executions and outcomes it compares, and what discrepancy it allows. This paper develops records that certify agreement at a declared resolution, rather than treating preservation alone as an agreement guarantee.

I. FROM PRESERVATION TO A GUARANTEE

  • Execution records must distinguish the logical task from its compiled and measured realizations, whose layout, routing, noise, and measurement conditions can change outcome laws.Valid stochastic runs may differ sample by sample even when governed by the same law.Source: §I · p. 1
  • Existing provenance systems capture research objects and quantum execution context, while reproducibility work measures agreement and accreditation checks accuracy against an ideal target.The paper's agreement criterion concerns implementations sharing a retained record; agreement alone does not establish accuracy.Source: §I · p. 1
  • The framework certifies retention using finite outcome-law comparisons or a continuous channel model, where latent noise flags and conditional quantum fidelity yield a finite precision budget without exhaustive outcome simulation.It applies information-theoretic and discernibility tools to execution contexts and tolerance-violating outcome-law distinctions.Source: §I · pp. 1–2
  • At Hellinger distance 0.0514011, two QAOA-1 executions sharing layout, gate-noise strength, and readout error exceed the 0.05 tolerance, so routing or angle bias must also be retained.The executions differ in routing and angle bias (0 versus 0.08 radians).Source: §I · p. 2

II. WHAT A SUFFICIENT RECORD PRESERVES

A retained record groups executions it cannot distinguish, and its risk is the largest outcome-law distance hidden within any such group. Sufficiency is therefore relative to a declared execution class, comparison metric, and tolerance.

II. WHAT A SUFFICIENT RECORD PRESERVES

  • The comparison object is the single-shot outcome law for each admissible complete context, not a finite-shot histogram; repeated runs governed by the same law can return different outcomes.The task, model and software definitions, generated circuit, and outcome data are retained separately as evidence.Source: §II · pp. 2–3
  • Risk is the largest Hellinger distance between outcome laws in any record cell, measuring the worst discrepancy hidden by identical retained fields.Record equality defines the partition, while closeness within tolerance need not be transitive.Source: §II · p. 2
  • The metric can be replaced by another declared metric or application-specific pseudometric, while Hellinger distance supports the metric and processing properties used in the framework.Its local geometry is the Fisher information of the outcome law.Source: §II · p. 2
  • A field set is sufficient when its risk is at most ε; a minimum-cost sufficient set minimizes declared positive field costs.Among tied minimum-cost sets, the selection uses smallest risk and then a fixed enumeration order.Source: §II · p. 2
  • For N independent shots, affinity tensorization gives an exact link between one-shot tolerance and future-transcript tolerance; correlated or drifting shots require a different transcript model.The future comparison length N is distinct from the sample size used to estimate a law.Source: §II · p. 2
  • The criterion is operational retention, not statistical sufficiency for an unknown parameter or a smallest-record claim for each individual execution.More retained fields cannot increase risk, whereas enlarging the admissible class can increase the risk of a fixed policy.Source: §II · pp. 2–3

III. CONTINUOUS PRECISION FROM QUANTUM CHANNELS

A channel-derived certificate bounds outcome-law differences throughout continuous record cells by combining stochastic noise-flag affinity with conditional quantum fidelity. It yields finite precision budgets without simulating outcome distributions, with scope and conservatism set by the declared circuit model.

III. CONTINUOUS PRECISION FROM QUANTUM CHANNELS

  • Continuous fields are recorded as declared bin labels whose widths must support the promised outcome-law diameter; the channel certificate gives a sufficient envelope, not a necessity claim about measured outputs.It avoids a sampled sensitivity envelope, grid-coverage assumption, and outcome simulation.Source: §III · pp. 3–4
  • The joint certificate combines the affinity of independent latent noise flags with conditional quantum fidelity for coherent rotations, and an observable-flag construction attains the bound.The bound is independent of register dimension and common programmed angles.Source: §III · p. 3
  • Independent stochastic contributions combine quadratically, while a shared coherent bias can accumulate across rotations before squaring; their contributions compose in the squared bound.Summing a separate Hellinger allowance for every noisy gate loses this structure.Source: §III · p. 3
  • 13 bits suffice for QAOA-1 with B < 0.049716, and the joint construction uses 13–15 bits across four circuits.The QAOA-1 allocation is (29, 8, 4), with 5,568 record labels; the certificate is uniform in programmed angles when the logical task is fixed.Source: §III · p. 4
  • At τ = 0.5, QAOA-1 requires 13 bits for N = 100 and 18 bits for N = 1000; the achievable channel-envelope rate is 2 log2 N + O(1), not a lower bound for the measured circuit.Source: §III · p. 4
  • The uniform channel bound yields a direction-dependent Fisher upper bound throughout the model domain, while one-shot attainability of the Cramér–Rao bound is not assumed.Actual measurements can be less sensitive than the certificate envelope.Source: §III · p. 4
  • Seven labels suffice and are necessary for the QAOA-1 design grid at ε = 0.05, versus 81 labels for its selected four-field record.This finite-grid comparison is three versus seven fixed-length bits and does not lower-bound the continuous precision budget.Source: §III · p. 4
  • Named-field records can cost more than encodings that preserve only outcome-visible distinctions, so the appropriate representation depends on whether the goal is comparison, attribution, or replay.The encoding contrast concerns permitted representations, not quantum advantage.Source: §III · p. 4

IV. FIELD RETENTION AND SEPARATION CERTIFICATES

Local ablations and single-field checks can miss violations concealed when several fields are omitted together. The separation certificate instead characterizes sufficient records by whether they distinguish every pair whose outcome laws exceed the tolerance.

IV. FIELD RETENTION AND SEPARATION CERTIFICATES

  • At tolerance 0.1, both fields are necessary although either single-field ablation has zero deficit and forgetting both yields H = 0.0893; the empty record hides a pair at distance 0.5412.Averaging over missing fields can conceal a failure rather than certify its absence.Source: §IV · p. 4
  • Single-field checks with sk ≤ ε for every omitted field do not guarantee RΩ(Bε) ≤ ε when those fields are omitted jointly.On Cartesian support, record cells are boxes but their images in outcome-law space can be curved; routing is discrete.Source: §IV · p. 4
  • A set S is ε-sufficient exactly when it intersects the distinguishing fields of every pair whose Hellinger distance exceeds ε, yielding a weighted hitting-set problem.The certificate directly checks whether every tolerance-violating pair is separated.Source: §IV · p. 4
  • For an inclusion-minimal sufficient set, each retained field has a violating pair that becomes indistinguishable when that field is removed; this does not establish global cost optimality.Global optimality requires solving the optimization or excluding every cheaper candidate; the small study enumerates all field sets.Source: §IV · pp. 4–5
  • Table I gives conservative channel-budget precision over the continuous noise box, not the unknown exact outcome-law diameter; its certificates hold for arbitrary programmed angles with the logical task fixed.Figure 1 compares joint, separate, and gatewise budgets and shows calculated transcript allocations, not measured hardware costs.Source: Table I · p. 5, Figure 1 · p. 5

V. AN EXECUTABLE FINITE-CLASS STUDY

The finite-class study tests field-retention policies against Hellinger-distance guarantees on simulated circuit contexts, then extends the analysis across archived task instances. For 28 archived QAOA instances, the unique minimum-cost sufficient set is {L, G, M, A} at ε = 0.05.

Study design

  • The study computes raw outcome laws by density-matrix propagation across four-qubit circuit instances and evaluates all 64 field sets with unit costs.The instances cover two hardware-efficient ansatz circuits and two four-cycle MaxCut QAOA circuits; the tolerance sweep is exploratory, not pre-registered.Source: §V · p. 5

Retention-policy comparison

  • Across the finite tolerance grid, the reference-mixture rule violates its guarantee throughout, while the singleton-threshold rule can still exceed ε when omissions combine.For QAOA-1 at ε = 0.05, the singleton rule reaches 0.0514; Figure 2 shows the policy comparison across tolerances.Source: §V · p. 5, Figure 2 · p. 6
  • The reference-mixture rule leaves diameters of 0.1215–0.2924, retaining only readout for ansatz instances and only gate noise for QAOA.For HEA-1, routing has zero local score at the identity layout but a single-field worst-case distance of 0.1211 elsewhere.Source: §V · p. 6, Table II · p. 6

Compositional failure

  • For QAOA-1 at ε = 0.05, omitting R or A alone gives risks 0.0416650 and 0.0496812, but omitting both gives 0.0514011.Both {L, R, G, M} and {L, G, M, A} are sufficient; the residual-risk tie-break selects the latter.Source: §V · pp. 6–7

Comparison target

  • The comparison target changes the retention conclusion: for HEA-1, retaining A meets a Z0-expectation tolerance of 0.03 but leaves distributional risk 0.2759.The passage distinguishes these targets and units rather than treating them as interchangeable criteria.Source: §V · p. 7

Transfer to archived instances

  • The residual-risk tie-break passes all thirteen archived QAOA instances with maximum risk 0.0444, but the additional instances are a retrospective comparison, not untouched validation.The tie-break was formulated after the first failures were observed, although its numerical selection uses only design-instance data.Source: §V · p. 7

Family certificate

  • For 28 archived QAOA instances, {L, G, M, A} is the unique minimum-cost sufficient set at ε = 0.05, with family risk 0.0444 at QAOA-1 seed 51.The QAOA-2 instance alone has singleton violations in L, G, M, and A; the HEA family requires all five law-affecting fields.Source: §V · p. 7

VI. FINITE-SHOT CERTIFICATION AND SCOPE

Finite-shot certification combines simultaneous uncertainty bounds with within-record risk tests, while the continuous channel certificate avoids sampled output laws when its model and parameter ranges are justified. The guarantees remain bounded by their declared contexts and assumptions.

Finite-shot certification

  • Simultaneous law-estimate bounds yield lower and upper risk bounds within record cells, classifying policies as passing, failing, or unresolved.A pass requires the upper bound to be at most ε; a failure requires the lower bound to exceed ε, and unresolved field deletions establish neither necessity nor minimality.Source: §VI · p. 7
  • 0.014 is the empirical Hellinger radius at 10^5 shots per context, versus 0.004 at 10^6; a radius of 0.025 requires 30,669 shots.For B = 16, M = 324, and α = 0.05, the elementary Hoeffding construction gives radii 0.250 and 0.141 at those shot counts and needs 109 shots for radius 0.025.Source: §VI · p. 7

Scope

  • The finite-shot performance claim is limited to uncertainty in estimated laws; the exact-law study makes no finite-shot performance claim.For extension to a larger context class, each context needs a same-record anchor whose law is within η, giving risk at most RΩ(S) + 2η; parameter checks alone cannot establish universal coverage.Source: §VI · pp. 7–8
  • The margin ε − RΩ(S) sets robustness to law perturbations, while the analytic channel certificate needs no sampled output laws once its model and parameter ranges are justified.A sufficient planning condition is 4η ≤ ε − R when planning from a known model risk R.Source: §VI · pp. 7–8

Finite-shot certification

  • 4.42 × 10^6 shots per context suffice for the residual-risk policy, versus 3.02 × 10^9 for the enumeration-order tie.These are conservative planning bounds for 324 contexts, not observed hardware costs or necessary sample complexities; the margin improves the sufficient budget by about 683-fold.Source: §VI · p. 8

Scope

  • Agreement does not establish accuracy against an ideal answer, nor does a retention certificate decompose experimental uncertainty.Distributional sufficiency also controls a declared bounded observable on the common outcome alphabet.Source: §VI · p. 8

VII. A FINITE-CONTEXT HARDWARE CONFIRMATION

Fresh six-qubit processor measurements compare raw outcome laws with laws after a fixed decoder at a common tolerance. On the tested finite context class, the decoder changes the minimum label budget; the result is conditional on the stated sampling assumptions.

Task and setup

  • A six-variable conditional boundary-refinement task from a synthetic CT formulation used a fixed decoder retaining boundary variables d(x) = (x2, x3).The raw and decoded alphabets have 64 and four outcomes; the guarantee concerns the two retained labels, not the entire segmentation.Source: §VII · p. 8
  • The confirmation used two layouts on a six-qubit processor, with 32,768 application shots in each of six contexts and three fresh jobs.The task, decoder, angles, and submitted native circuit bytes were unchanged; scout shots were excluded.Source: §VII · p. 8

Minimum field costs

  • All three prespecified targets hold, yielding minimum sufficient field costs of zero for decoded laws and one for raw laws.The raw empty record is excluded by a lower-bound witness; backend and compiler are constant in this finite class.Source: §VII · p. 8, Table IV · p. 9

Minimum label counts

  • Two labels are necessary and sufficient for raw laws, while one suffices for decoded laws at ε = 0.15.The raw empty record fails, the raw layout record passes, and the decoded empty record passes; these conclusions hold on the common confidence event.Source: §VII · pp. 8–9, Table IV · p. 9

Scope

  • The tolerance is a distribution-comparison criterion, not a validated segmentation-error threshold; results remain conditional on stationary independent shots.Basis-state controls were consistent with the raw/decoded contrast but insufficient to identify a causal noise mechanism; the experiment does not validate the continuous channel envelope.Source: §VII · p. 9

VIII. DISCUSSION

The discussion separates finite-context field retention from continuous channel-based precision bounds. It emphasizes that agreement guarantees depend on the declared model and evidence class, and do not replace the execution archive.

Continuous precision certificate

  • The continuous certificate bounds uncertainty within record cells by composing independent stochastic-channel flag affinities with coherent-rotation fidelity.Comparisons hold the logical task fixed; finite-class witnesses and the channel certificate answer complementary questions.Source: §VIII · p. 9

Model assumptions

  • The product-formula certificate requires independent mixture flags for the declared stochastic channels and inclusion of all coherent differences in the rotation budget.Unmodelled correlated environments, context-dependent measurement, or compiler changes can invalidate it.Source: §VIII · p. 9

Scope

  • A calibrated parameter's uncertainty interval must be covered alongside recording quantization; the certificate does not establish accuracy against an ideal answer.The construction also does not justify deleting the evidential archive.Source: §VIII · p. 9
  • The analytic bound is independent of Hilbert-space dimension once the channel inventory is supplied, but may overestimate sensitivity of a particular measurement.Finite-class law generation, pair enumeration, and subset optimization remain costly and separate from scalable evaluation of a proposed channel budget.Source: §VIII · p. 9

Hardware comparison

  • The hardware result shows a decoder can change the minimum comparison-label budget at one tolerance, conditional on stationary independent shots and the tested finite context class.It does not answer whether an independently justified channel envelope leaves sufficient acquisition margin under measured drift.Source: §VIII · p. 9

Representation and archive

  • Named-field records may cost more than encodings that preserve only distinctions visible in outcome laws; the appropriate representation depends on whether the goal is comparison, attribution, or replay.Sufficient comparison records do not replace the detailed execution archive needed to investigate how a result was produced.Source: §VIII · p. 9
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