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Phase-cycled randomized benchmarking of quantum processors: recovering hidden classical noise correlations

Mirza Samad Ahmed Baig, Syeda Anshrah Gillani, Abdul Akbar Khan, Muhammad Omer Khan

arXiv:2609.06448v1cs.ETcs.CLcs.DC

TL;DR

RB averages can hide classical temporal correlations, and dense idle-duration scans may not identify whether evolution continues across slot boundaries. The paper proves a continuous/reset equivalence for symmetric telegraph noise and constructs a phase-cycle covariance measurement; two hardware acquisitions report positive correlated-minus-reset contrasts while negative-control intervals include zero.

  • Problem

    Which feature of the Clifford average hides a symmetric fluctuator, and can changing the measured observable recover the information it removes?

    Source: §I · p. 1

  • Method

    The paper establishes a symmetric-telegraph reset equivalence and uses phase cycling to estimate a connected sine-phase covariance, with paired-block bootstrap intervals and a separate conservative confidence set.

    Source: A. Closed response for continuous… · p. 3, §IV · p. 3, §IX · p. 7

  • Results

    The primary correlated-minus-reset contrasts were 0.2540 (nominal 95% interval [0.1775, 0.3310]) and 0.2114 (nominal 95% interval [0.1359, 0.2851]); all individual negative-control intervals contained zero.

    Source: B. Injected phase-cycle results · p. 5

  • Takeaways & Limitations

    The work connects an RB mean-response identifiability limit to a controlled measurement of engineered inter-slot dependence.

    Source: §IX · p. 7

    The stated scope is classical idle dephasing with Clifford-independent phases; finite gate durations, gate-dependent error, leakage, nonstationary SPAM, or unmatched single-slot marginals can invalidate the simple interpretation.

    Source: §VIII · p. 7

Abstract

from arXiv · show

Randomized benchmarking can hide classical temporal correlations because its Clifford-twirled response is even in the noise phase. For a stationary symmetric telegraph fluctuator, we show that continuous evolution and independent stationary resets at slot boundaries yield identical mean responses for arbitrary fixed idle modulations. We construct an eight-setting phase-cycle measurement of the connected sine-phase covariance under ideal Clifford twirling and classical idle dephasing. This observable vanishes for independent slot noise and fixed detuning without a weak-phase or Gaussian approximation. A closed telegraph response, independent circuit calculations and 800 simulation trials validate the construction and quantify the empirical coverage of a paired bootstrap estimator. A separate conservative confidence set states its finite-sample assumptions. Two acquisitions on an IBM processor compare engineered shared-sign and independently reset phases with identical marginals. Their primary contrasts are 0.254 and 0.211, with empirical 95% intervals [0.177, 0.331] and [0.136, 0.285], respectively; all negative-control intervals include zero. At equal shot and sensingwindow budgets, an ideal Ramsey/echo estimator is more precise in every tested class. The result supplies an explicit connection between a benchmarking identifiability limitation and a controlled correlation measurement. No native or quantum-memory detection is claimed.

I. INTRODUCTION

An averaged randomized-benchmarking decay does not establish memoryless noise: classical environments can retain information while producing the same decay. The section distinguishes what idle-duration scans reveal from whether evolution is continuous across slots, and introduces a phase-cycle measurement to address that identifiability gap.

I. INTRODUCTION

  • A single-exponential RB response does not establish memoryless noise, and a fitted average error is not a worst-case error certificate.Classical environments can preserve information across gates while producing the same averaged decay.Source: §I · p. 1
  • Idle scans and engineered-noise RB tests already probe dephasing and correlations, so neither inserting an idle window nor calibrating against injected correlations is claimed as new.The proposed contribution is the explicit Clifford-twirled connected sine covariance and its matched-marginal test, not RB blindness or sine-correlation spectroscopy itself.Source: §I · p. 1
  • The paper separates within-slot phase-distribution spectroscopy from identifying continuous evolution versus independent slot resets; even a dense gap scan may not distinguish the latter.Source: §I · p. 1

II. WHAT A GAP SCAN CAN IDENTIFY

  • For a single slot, Clifford averaging makes the normalized contrast a product of mean even phase factors, so the scan probes phase statistics through their even response.The phase variance for exponential field autocorrelation is shared by Gaussian and symmetric telegraph noise, despite their different complete phase distributions.Source: §II · p. 2
  • The fitted local EPC exponent is an effective finite-interval description: its offset must be fitted, and a quadratic response can also arise from deterministic frequency error.Source: §II · p. 2

A. An exact reset equivalence

For a stationary symmetric two-state telegraph fluctuator, continuous evolution and independent stationary resets at slot boundaries give identical mean RB contrasts. This equivalence holds across unequal slot durations and arbitrary fixed slot modulations, but is limited to the stated model.

A. An exact reset equivalence

  • Continuous telegraph evolution and independent stationary slot resets yield the same mean RB contrasts, including for unequal idle durations and arbitrary fixed slot modulations.Source: A. An exact reset equivalence · p. 2
  • The equivalence means no scan of ideal mean contrasts over gap durations distinguishes this continuous/reset pair; the reset model can retain within-slot correlations.It is memoryless across slot boundaries, not necessarily a continuous-time Lindblad semigroup.Source: A. An exact reset equivalence · p. 2
  • The proof assumes one symmetric two-state fluctuator and gate-independent coupling; multiple fluctuators, unequal switching rates, or environment-dependent gate errors need not obey it.An even modulation of the same scalar phase cannot break the equivalence; changing response parity can.Source: A. An exact reset equivalence · p. 2

III. PHASE CYCLING RECOVERS THE HIDDEN CORRELATOR

Controlled phase cycling changes the Clifford-twirled response so that a connected sine-phase covariance can reveal inter-slot dependence. The witness rejects independent phases and fixed detuning under the stated model, but it is sufficient rather than universal.

III. PHASE CYCLING RECOVERS THE HIDDEN CORRELATOR

  • For two slots, the measurement combines phase-cycle joint responses with matched single-slot marginals to form the connected covariance.Stationarity or engineered replay must ensure that single-slot laws match the corresponding two-slot marginals.Source: §III · pp. 2–3
  • The connected sine-phase covariance responds to inter-slot dependence without a Gaussian or small-angle approximation, while fixed detuning and independent resets give zero.The connected subtraction is essential because fixed detuning can produce nonzero marginal sine responses.Source: §III · p. 3
  • A nonzero covariance excludes independence only within the stated measurement model; a revival under the controlled bias is not evidence of quantum memory.At β = π/2 the coefficient is maximized, but the setting introduces a substantial known coherent rotation and is not an ordinary low-error EPC measurement.Source: §III · p. 3
  • The witness is sufficient but incomplete: correlated phase distributions can have zero sine covariance, including near integer multiples of π.Varying idle duration or coupling can address particular zeros but does not make this second-moment witness a universal dependence test.Source: §III · p. 3

A. Closed response for continuous telegraph noise

The closed response relates the connected sine-phase covariance to telegraph evolution across two windows and an uncoupled separation. The estimation section pairs a block-bootstrap interval with a separate conservative confidence set and states the assumptions behind each.

A. Closed response for continuous telegraph noise

  • At fixed window duration, the closed response’s dependence on uncoupled separation gives a model prediction for the inverse correlation time, not a hardware measurement.A physical-qubit delay usually remains coupled to its environment, so realizing an uncoupled separation requires separate control and error characterization.Source: A. Closed response for continuous… · p. 3

IV. ESTIMATION AND FINITE-SAMPLE SCOPE

  • The estimator uses four two-slot phase settings and four marginal settings to form block quantities J, U, and V.Distinct blocks make products of marginal means unbiased while allowing arbitrary dependence within a block, avoiding the finite-sample covariance bias of multiplying paired sample means.Source: §IV · p. 3
  • The practical interval resamples whole blocks and is an empirical percentile-bootstrap approximation, not a finite-sample coverage theorem.The held-out calibration table reports percentages over 200 trials per class; detection means the nominal two-sided interval excludes zero.Source: §IV · p. 3, Table I · p. 4
  • For independent identically distributed blocks, bounded block quantities support a separate conservative confidence set with at least 1 − α coverage for Cβ.It bounds each population mean using Hoeffding’s inequality and a union bound, then evaluates the minimum and maximum of j − uv over the resulting rectangular set.Source: §IV · p. 3
  • Independent blocks are substantive: slowly drifting native environments can violate the assumption, requiring time ordering, suitable block lengths, or independent acquisitions.The conservative bound is for ideal bounded observations and is not asserted unchanged after hardware readout-gain correction.Source: §IV · p. 3

V. NUMERICAL VALIDATION

Independent calculations validate the reset equivalence, closed response, and random-Clifford product-twirl identity to near machine precision. Simulations then assess the response and interval behavior across telegraph and control classes.

V. NUMERICAL VALIDATION

  • The maximum reset-equivalence discrepancy across 19 correlation times, three idle durations, and circuit lengths m = 0, . . . , 50 is 4.66 × 10^-15.The closed form agrees with eight transfer-matrix means to 1.39 × 10^-16, and all 242 Clifford pairs match the product-twirl identity to 3.33 × 10^-16 for weak and strong test angles.Source: §V · p. 4
  • The sampling study uses 200 independent trials per class, each with 512 independent blocks and 64 binomial shots per setting.For the continuous telegraph model, σ = 1.6 rad/µs, τc = 2 µs, δ = 0.5 µs, h = 0.1 µs, and exact RTN covariance is 0.1704.Source: §V · p. 4
  • Simulation controls include independent resets with the same complete single-slot phase law, deterministic phase, and independent Gaussian slot phases.The study reports Table I results and notes that empirical coverage variation makes the distinction between empirical and guaranteed intervals material.Source: §V · p. 4

A. Additional assumption checks

Additional numerical checks test unequal-slot modulations, nonsymmetric phase laws, and cycle angles. An ideal resource-matched Ramsey/echo comparison estimates the same sine covariance and is more precise in every tested class.

A. Additional assumption checks

  • Across 200 unequal-slot schedules with piecewise real modulation of either sign, continuous and reset means differ by less than 10^-14.Tests of 100 discrete nonsymmetric phase laws and varied cycle angles also verify Eq. (7) below the same tolerance; a deliberately asymmetric telegraph example instead gives discrepancy 0.000852.Source: A. Additional assumption checks · p. 4

B. An ideal resource-matched Ramsey comparator

  • The four-setting Ramsey/echo construction combines cosine responses for phase sums and differences with two single-slot sine means to obtain the sine covariance.It is motivated by established correlation spectroscopy, but is not a reproduction of the RESOLUTE experiment or its full control sequence.Source: B. An ideal resource-matched Ramsey… · p. 4
  • The ideal resource-matched Ramsey/echo estimate has lower root mean squared error than the RB estimate in every tested noise class.Both estimate the same sine covariance, but the comparison excludes native drift and imperfect gates and matches shot and sensing-window budgets rather than gate duration or wall time.Source: B. An ideal resource-matched Ramsey… · p. 4, Table II · p. 4

VI. HARDWARE CONTROLS

The phase-cycle experiment compares engineered shared-sign and independently reset phases with identical single-slot marginals. Across two acquisitions, the primary contrast was positive while all negative-control intervals included zero; these results calibrate engineered dependence, not native hardware memory.

A. A test that holds the slot marginals fixed

  • The four-arm cycle compares shared random signs, independent signs with identical single-slot distributions, deterministic phases, and no injected phase.For θ = 0.8 rad, the ideal connected response is (4/9) sin2 θ for shared signs and zero for each control; ordinary unbiased RB means coincide for the first three arms.Source: A. A test that holds the slot… · p. 4
  • Each block uses eight settings with 320 ns idles; virtual phase and bias rotations are protected by barriers and remain outside the Clifford inverse.The processor was ibm fez q46; setting and template order were randomized, which reduces order confounding but does not prove stationarity.Source: A. A test that holds the slot… · p. 5
  • The primary comparison was prespecified, and injected phases stayed fixed within each circuit’s shot group rather than being replayed across circuits.The acquisition calibrates recovery of engineered inter-slot dependence, not detection of a native hardware memory process.Source: A. A test that holds the slot… · p. 5

B. Injected phase-cycle results

  • The correlated-minus-reset contrasts were 0.2540 [0.1775, 0.3310] and 0.2114 [0.1359, 0.2851]; all negative-control intervals included zero.Table III reports the measured contrasts and ideal population values; Fig. 2 shows acquisition estimates, paired bootstrap intervals, and ideal predictions.Source: Table III · p. 5, B. Injected phase-cycle results · p. 5, Figure 2 · p. 6
  • The two completed acquisitions used the same qubit on the same day, testing within-session repeatability rather than robustness across devices or calibration cycles.Three unsuccessful jobs had no usable counts and were excluded; the completed results do not establish native memory or a physical correlation-time scan.Source: B. Injected phase-cycle results · pp. 5–6

C. Readout calibration sensitivity

A readout-gain sensitivity analysis checks whether the positive hardware point estimates depend on treating unobserved calibration errors as zero. Complementary idle-gap analyses report calibration observations and identify limitations of the legacy analysis and repeated-sequence diagnostic.

C. Readout calibration sensitivity

  • The calibration-only contrast ranges were [0.2536, 0.2590] for acquisition 1 and [0.2093, 0.2171] for acquisition 2.These ranges do not include experimental sampling error and are not confidence intervals for the primary contrast; they do not test drift or setting-dependent SPAM.Source: C. Readout calibration sensitivity · p. 6

D. Complementary idle-gap calibration

  • The legacy idle-gap campaign’s effective exponents were 1.00 ± 0.10, 1.51 ± 0.08, and 1.72 ± 0.15 for fast, crossover, and quasi-static arms.The largest discrepancy from the stored response prediction was 1.61 standard errors; these are calibration observations for Eq. (3), not independent evidence of priority or universal classifier performance.Source: D. Complementary idle-gap calibration · p. 6
  • The legacy EPC subtraction is only leading-order, its data-dependent cuts and separately estimated uncertainties lack established interval coverage, and its Hahn control does not validate finite-time within-slot switching.The baseline-subtracted EPC misses exact cancellation because independent depolarizing contractions compose multiplicatively; the echo control refocuses programmed phase, not the full finite-time telegraph response.Source: D. Complementary idle-gap calibration · p. 6
  • Repeated-sequence variance probes changes between circuit executions, a different timescale from memory between gates, and is a diagnostic rather than a necessary condition for inter-gate memory.The injected quasi-static arm gave W_rep = 4.02, with native controls near unity; averaging many independent trajectories may suppress the statistic.Source: D. Complementary idle-gap calibration · p. 6

VII. NATIVE SURVEY AND LIMITS OF A QUANTITATIVE NULL

The native screening candidates did not reproduce on follow-up, while a stored idle-gap fit was assessed against a proposed correlated-noise model family. The resulting diagnostics do not support an unconditional 95% exclusion of correlated noise.

VII. NATIVE SURVEY AND LIMITS OF A QUANTITATIVE NULL

  • The two gap-exponent candidates did not reproduce: q53 on ibm fez changed from 1.87 ± 0.22 to 1.26 ± 0.14, and q154 on ibm marrakesh from 2.63 ± 0.54 to 1.56 ± 0.28.The screen covered 16 qubits; selection effects and temporal drift are both possible explanations, and the data do not uniquely attribute the changes to either.Source: §VII · p. 6
  • With the stored diagonal EPC error model, minimum residual χ2 ranged from 15.81 to 18.97 across 13 tested correlation times (20–200,000 ns), and all exceeded 12.59, the 95th percentile of χ2.At every proposed fraction and correlation time, both nuisance parameters were refit using weighted nonnegative least squares.Source: §VII · p. 7
  • An empty intersection on the tested grid signals model incompatibility, not exclusion of every correlated contribution; the covariance is estimated and omits unavailable cross-gap covariance.The authors therefore do not quote an unconditional 95% correlated-noise exclusion; raw-block reanalysis, adequacy checks, and coverage calibration are required for a quantitative exclusion.Source: §VII · p. 7

VIII. DISCUSSION

The discussion connects reset equivalence with phase cycling as a way to recover a hidden inter-slot correlation, while emphasizing the method’s classical-dephasing scope and limits. The conclusion distinguishes this operational memory observable from stronger claims the data do not support.

VIII. DISCUSSION

  • The connected subtraction separates shared random phase from deterministic frequency shift, while the reset control tests specificity at fixed single-slot statistics.Thus duration-dependent noise measurements can leave inter-slot dependence unidentified; phase cycling changes the observable rather than merely refining the same even response.Source: §VIII · p. 7
  • The interpretation is limited to classical idle dephasing with Clifford-independent phases; gate-dependent error, leakage, nonstationary SPAM, or unmatched marginals can invalidate it.Fixed-detuning and no-injection controls do not establish absence of these effects under all operating conditions; a physical lag scan needs additional control validation.Source: §VIII · p. 7
  • At equal shot and sensing-window budgets, ideal Ramsey/echo estimation is more precise in every tested noise class, so this method shows no precision advantage.The method also does not inherit ordinary RB’s fitted separation of gate decay and arbitrary SPAM coefficients.Source: §VIII · p. 7
  • The conservative mean-rectangle confidence set has stated coverage under independent bounded blocks, whereas bootstrap coverage is empirical.Neither assessment implies quantum-memory certification or a worst-case gate-error bound.Source: §VIII · p. 7

IX. CONCLUSION

  • A controlled phase cycle recovers the sine-phase covariance hidden by ordinary even Clifford response, with independent-noise and deterministic-detuning nulls.The construction provides empirical and conservative uncertainty assessments; the hardware controls and native-data audit distinguish the observable from stronger unsupported claims.Source: §IX · pp. 7–8
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