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arxiv: 2507.21883 · v3 · submitted 2025-07-29 · 🪐 quant-ph

Sampling (noisy) quantum circuits through randomized rounding

Victor Martinez , Omar Fawzi , Daniel Stilck Fran\c{c}a This is my paper

Pith reviewed 2026-05-19 02:21 UTC · model grok-4.3

classification 🪐 quant-ph
keywords randomized roundingnoisy quantum circuitsMax-Cutcombinatorial optimizationquantum simulationapproximation ratiodepolarizing noise
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0 comments X p. Extension

The pith

A classical Gaussian randomized rounding procedure applied to the two-qubit marginals of a noisy quantum circuit produces bit-string samples whose expected cost closely matches the noisy device's performance on problems like Max-Cut.

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

The paper establishes that for combinatorial problems whose cost depends only on pairwise correlations, such as Max-Cut, the two-qubit reduced states from a noisy quantum circuit suffice to generate classical samples via Gaussian randomized rounding. These samples achieve an expected approximation ratio of 1 minus a term that scales as O of (1 minus p) to the power D for a depth-D circuit under local depolarizing noise of strength p. If this holds, near-term noisy quantum hardware for such tasks admits an efficient classical surrogate that reproduces both the average performance and the full energy distribution of the circuit. The approach extends to other noise models and is validated through large-scale simulations plus experiments on IBMQ devices. This supplies a concrete benchmark for assessing when error mitigation or fault tolerance would be needed to exceed classical performance on these problems.

Core claim

For optimization problems whose objective depends only on two-body correlations, Gaussian randomized rounding applied to the two-qubit marginals of a depth-D circuit subject to local depolarizing noise p yields a classical distribution over bit strings whose expected cost achieves an approximation ratio of 1-O[(1-p)^D] relative to the noisy quantum circuit itself.

What carries the argument

Gaussian randomized rounding applied to the two-qubit marginals of the noisy circuit

If this is right

  • For any depth-D circuit under local depolarizing noise p the classical sampler matches the noisy quantum expected cost up to a 1-O[(1-p)^D] factor on Max-Cut.
  • The rounded samples reproduce the full energy distribution of the noisy circuit in numerical simulations.
  • The same rounding procedure exhibits similar fidelity under noise models other than local depolarizing.
  • The method supplies an efficient classical surrogate that can be used to benchmark future error-mitigated or fault-tolerant quantum optimization runs.

Where Pith is reading between the lines

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

  • The result implies that any potential quantum advantage for noisy two-body optimization tasks is confined to regimes where (1-p)^D remains appreciable.
  • Extending the technique to objectives that involve higher-order correlations would require rounding procedures that capture multi-qubit marginals.
  • Classical approximation algorithms for Max-Cut could be hybridized with the quantum marginals to produce improved rounding schemes.

Load-bearing premise

The objective function depends only on two-body correlations such as in Max-Cut, allowing the expected cost to be determined from the circuit's two-qubit marginals alone.

What would settle it

For a fixed depth-D circuit and noise strength p, generate noisy samples from the quantum device on a Max-Cut instance, compute their empirical average cost, then compare it to the average cost of an equal number of rounded classical samples; a statistically significant gap larger than the predicted O[(1-p)^D] term would falsify the approximation guarantee.

Figures

Figures reproduced from arXiv: 2507.21883 by Daniel Stilck Fran\c{c}a, Omar Fawzi, Victor Martinez.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p026_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p027_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗
read the original abstract

The present era of quantum processors with hundreds to thousands of noisy qubits has sparked interest in understanding the computational power of these devices and how to leverage it to solve practically relevant problems. For applications that require estimating expectation values of observables the community developed a good understanding of how to simulate them classically and denoise them. Certain applications, like combinatorial optimization, however demand more than expectation values: the bit-strings themselves encode the candidate solutions. While recent impossibility and threshold results indicate that noisy samples alone rarely beat classical heuristics, we still lack classical methods to replicate those noisy samples beyond the setting of random quantum circuits. Focusing on problems whose objective depends only on two-body correlations such as Max-Cut, we show that Gaussian randomized rounding in the spirit of Goemans-Williamson applied to the circuit's two-qubit marginals-produces a distribution whose expected cost is provably close to that of the noisy quantum device. For instance, for Max-Cut problems we show that for any depth-D circuit affected by local depolarizing noise p, our sampler achieves an approximation ratio $1-O[(1-p)^D]$, giving ways to efficiently sample from a distribution that behaves similarly to the noisy circuit for the problem at hand. Beyond theory we run large-scale simulations and experiments on IBMQ hardware, confirming that the rounded samples faithfully reproduce the full energy distribution, and we show similar behaviour under other various noise models. Our results supply a simple classical surrogate for sampling noisy optimization circuits, clarify the realistic power of near-term hardware for combinatorial tasks, and provide a quantitative benchmark for future error-mitigated or fault-tolerant demonstrations of quantum advantage.

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

0 major / 2 minor

Summary. The paper introduces a classical sampling method for noisy quantum circuits applied to combinatorial optimization problems (e.g., Max-Cut) whose objective depends only on two-body correlations. Gaussian randomized rounding is applied to the two-qubit marginals of a depth-D circuit subject to local depolarizing noise of strength p, yielding a classical distribution whose expected cost approximates the noisy quantum expectation with ratio 1-O[(1-p)^D]. The theoretical guarantee is supported by large-scale simulations and IBMQ hardware experiments that show the rounded samples reproduce the full energy distribution, with similar behavior under other noise models.

Significance. If the central claim holds, the work supplies a simple, efficient classical surrogate for sampling noisy optimization circuits and quantifies the realistic power of near-term hardware for combinatorial tasks via an explicit, parameter-free approximation ratio derived from noise propagation and rounding properties. The large-scale simulations and IBMQ runs provide reproducible empirical support and a useful benchmark for error-mitigated or fault-tolerant demonstrations.

minor comments (2)
  1. [Theoretical derivation of approximation ratio] The proof of the approximation ratio (in the section deriving Theorem 1) would benefit from an explicit expansion of the rounding error term (e.g., the contribution from higher-order terms in the arcsin expansion) to make the hidden constant in the O[(1-p)^D] notation transparent.
  2. [Experimental results section] Figure captions for the energy-distribution comparisons should state the number of samples used and any statistical tests performed, to allow readers to assess the fidelity claim quantitatively.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our work, the clear summary of the central claims, and the recommendation for minor revision. We are pleased that the significance for providing a classical surrogate for noisy optimization circuits is recognized, along with the value of the simulations and hardware experiments.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives the approximation ratio 1-O[(1-p)^D] directly from the uniform decay bound on two-qubit correlations under local depolarizing noise of strength p per layer (which follows from the noise model and circuit depth) combined with the known properties of Gaussian randomized rounding applied to the Gram matrix of those marginals. The restriction to objectives depending only on two-body terms makes the expected cost exactly recoverable from the marginals by definition, but this is an explicit modeling choice rather than a hidden fit. No parameter is tuned to the target distribution, no self-citation supplies a load-bearing uniqueness theorem, and the rounding error is bounded by standard analysis of the arcsin or equivalent function without smuggling an ansatz. The result is therefore a genuine guarantee under the stated assumptions rather than a restatement of the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the objective is fully determined by two-body terms and on standard properties of Gaussian rounding and depolarizing noise channels.

axioms (1)
  • domain assumption The objective function depends only on two-body correlations
    This is invoked to reduce the expected cost to a function of the two-qubit marginals alone.

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