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arxiv: 2605.22909 · v1 · pith:43F6VQNVnew · submitted 2026-05-21 · 🪐 quant-ph

Sample-efficient benchmarking of shallow all-to-all random quantum circuits

Gregory Bentsen , Bill Fefferman , Soumik Ghosh , Michael J. Gullans , Yinchen Liu This is my paper

Pith reviewed 2026-05-25 05:53 UTC · model grok-4.3

classification 🪐 quant-ph
keywords random circuit samplingnonlinear cross-entropyheavy output generationdepolarizing noisesample efficiencyall-to-all quantum circuitsquantum benchmarkingNISQ devices
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The pith

Nonlinear cross-entropy separates noisy quantum computers from classical spoofers in shallow all-to-all random circuits.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that nonlinear cross-entropy supplies a sample-efficient benchmark for shallow-depth all-to-all random quantum circuits. Its score distinguishes noisy quantum devices from state-of-the-art classical spoofers even when depolarizing noise is present. The authors also introduce a binary classifier based on heavy output generation that requires only logarithmic samples at short depths. Existing linear cross-entropy benchmarks can be fooled by noise, so no reliable test existed for regimes where quantum sampling might still be classically hard. The evidence rests on exact formulas obtained via replica tricks for Brownian circuit ensembles together with numerical checks on discrete Haar-random circuits.

Core claim

The nonlinear cross-entropy provides a sample-efficient benchmark for shallow-depth all-to-all random quantum circuits whose score cleanly separates noisy quantum computers from state-of-the-art classical spoofers, even in the presence of depolarizing noise. A binary classifier based on heavy output generation features logarithmic sample complexity at short depth. Analytic expressions for all-to-all Brownian circuit ensembles derived using replica tricks support these results, and numerical simulations corroborate them for discrete Haar-random unitary circuits.

What carries the argument

Nonlinear cross-entropy, obtained from replica-trick calculations on Brownian circuit ensembles, which quantifies output separation under depolarizing noise and enables the heavy-output binary classifier.

If this is right

  • Shallow-depth regimes become verifiable even though linear cross-entropy is spoofable.
  • The heavy-output classifier decides quantum versus classical origin with only logarithmically many samples at short depth.
  • Depolarizing noise leaves a usable gap between quantum and classical score distributions.
  • Benchmarking can now reach circuit depths where sampling remains plausibly classically intractable.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same separation might appear in circuits with sparser connectivity if the noise model stays local.
  • Logarithmic sample complexity suggests the method could remain practical as qubit number grows.
  • Combining the classifier with existing error-mitigation techniques could extend its reach to higher noise rates.
  • Direct tests on current hardware would supply the next concrete check beyond the paper's numerics.

Load-bearing premise

The analytic expressions derived using replica tricks for all-to-all Brownian circuit ensembles accurately model and predict the separation under depolarizing noise for discrete Haar-random unitary circuits as well.

What would settle it

A numerical simulation or hardware experiment in which the nonlinear cross-entropy score achieved by a noisy quantum circuit falls inside the distribution reachable by state-of-the-art classical spoofers would falsify the claimed separation.

Figures

Figures reproduced from arXiv: 2605.22909 by Bill Fefferman, Gregory Bentsen, Michael J. Gullans, Soumik Ghosh, Yinchen Liu.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Numerical estimation of [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Numerical estimation of [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Numerical estimation of Var[ [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Numerical estimation of Var[ [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Numerical estimation of [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Ground-state pairings relevant to the spectrum of [PITH_FULL_IMAGE:figures/full_fig_p026_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Two categories of coupling coefficients [PITH_FULL_IMAGE:figures/full_fig_p027_12.png] view at source ↗
read the original abstract

Random circuit sampling (RCS) remains one of the most competitive frameworks for demonstrating quantum advantage in near-term noisy intermediate-scale quantum (NISQ) hardware. Unfortunately, absent error-correction, existing benchmarks to characterize these experiments, like linear cross-entropy, have been classically spoofed due to noise. Because of this, there are interesting regimes, like shallow-depth random quantum circuits, where sampling is plausibly classically intractable, but no existing benchmark can distinguish between a noisy quantum computer and an adversarial classical spoofer. In this paper, we demonstrate that the nonlinear cross-entropy provides a sample-efficient benchmark for shallow-depth all-to-all random quantum circuits whose score cleanly separates noisy quantum computers from state-of-the-art classical spoofers, even in the presence of depolarizing noise. Further, we develop a binary classifier based on the notion of heavy output generation that features logarithmic sample complexity at short depth. Our evidence comes from exact analytic expressions for all-to-all Brownian circuit ensembles derived using replica tricks, and numerical simulations that corroborate these results for discrete Haar-random unitary circuits.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript claims that the nonlinear cross-entropy provides a sample-efficient benchmark for shallow-depth all-to-all random quantum circuits whose score cleanly separates noisy quantum computers from state-of-the-art classical spoofers under depolarizing noise; it further introduces a heavy-output-generation binary classifier with logarithmic sample complexity at short depth. Evidence is drawn from exact analytic expressions obtained via replica tricks on all-to-all Brownian circuit ensembles together with numerical simulations that corroborate the results for discrete Haar-random unitary circuits.

Significance. If the claimed separation holds and the analytics transfer reliably, the work would supply a practical tool for characterizing NISQ random-circuit-sampling experiments in the shallow-depth regime where linear cross-entropy benchmarks are known to be classically spoofable, thereby strengthening experimental claims of quantum advantage.

major comments (2)
  1. [Abstract] Abstract: the central claim that replica-trick analytic expressions derived for continuous all-to-all Brownian ensembles accurately predict both the nonlinear cross-entropy score and the heavy-output classifier performance for discrete Haar-random unitaries under depolarizing noise is load-bearing, yet the manuscript provides no explicit error bounds, convergence analysis, or section detailing the modeling step that justifies the transfer from the continuous to the discrete finite-depth setting.
  2. [Numerical simulations (referenced in Abstract)] The numerical simulations are stated to corroborate the analytic results for discrete circuits, but without a dedicated section or table reporting quantitative agreement (e.g., deviation in the nonlinear cross-entropy value or classifier accuracy as a function of depth and system size), it is impossible to assess whether the numerics close the modeling gap or merely illustrate qualitative trends.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback. The two major comments identify gaps in the justification for transferring analytic results from the continuous Brownian ensemble to discrete Haar-random circuits and in the quantitative presentation of numerical corroboration. We address each point below and will incorporate revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that replica-trick analytic expressions derived for continuous all-to-all Brownian ensembles accurately predict both the nonlinear cross-entropy score and the heavy-output classifier performance for discrete Haar-random unitaries under depolarizing noise is load-bearing, yet the manuscript provides no explicit error bounds, convergence analysis, or section detailing the modeling step that justifies the transfer from the continuous to the discrete finite-depth setting.

    Authors: We agree that the manuscript would benefit from an explicit discussion of the modeling assumptions underlying the transfer. The Brownian ensemble is used because it permits exact replica-trick calculations while reproducing the leading-order statistics of Haar-random gates at short depth; the numerics are intended to confirm that the resulting expressions remain predictive for discrete circuits. In the revision we will add a dedicated subsection (likely in Section II) that (i) recalls the replica-trick derivation, (ii) states the regime of validity (small depth relative to system size, large number of gates per layer), and (iii) supplies numerical error bounds obtained by comparing the analytic formulas to exact small-system simulations. Convergence plots versus depth and qubit number will be included. revision: yes

  2. Referee: [Numerical simulations (referenced in Abstract)] The numerical simulations are stated to corroborate the analytic results for discrete circuits, but without a dedicated section or table reporting quantitative agreement (e.g., deviation in the nonlinear cross-entropy value or classifier accuracy as a function of depth and system size), it is impossible to assess whether the numerics close the modeling gap or merely illustrate qualitative trends.

    Authors: We concur that quantitative metrics are needed. The current figures show visual agreement but do not tabulate deviations. In the revised manuscript we will add an appendix (or new subsection) containing a table that reports, for depths d = 1…10 and qubit numbers n = 4…12, the mean absolute deviation and relative error between the analytic Brownian predictions and the Monte-Carlo estimates for both the nonlinear cross-entropy and the heavy-output classifier accuracy. Statistical uncertainties from finite sampling will also be listed, allowing readers to judge the size of the modeling gap directly. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives exact analytic expressions for all-to-all Brownian circuit ensembles via replica tricks and uses numerical simulations to corroborate applicability to discrete Haar-random unitaries under depolarizing noise. No load-bearing step reduces by construction to a fitted input, self-definition, or self-citation chain; the replica-trick derivation and separate numerical check constitute independent content against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities used in the derivations or simulations.

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