Sampling (noisy) quantum circuits through randomized rounding

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 a recovery ratio <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>1</mml:mn> <mml:mo>&amp;#x2212;</mml:mo> <mml:mi>O</mml:mi> <mml:mo stretchy="false">[</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>&amp;#x2212;</mml:mo> <mml:mi>p</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>D</mml:mi> </mml:msup> <mml:mo stretchy="false">]</mml:mo> </mml:math> , 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.