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 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.
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 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.
As quantum devices grow in size while remaining affected by noise, a key challenge is to determine when and how they can surpass classical computation on practical tasks. This requires the development of highly efficient classical simulation algorithms. Pauli backpropagation algorithms are among the most promising, having shown efficiency in simulating certain classes of parameterized quantum circuits, strong candidates for near‑term quantum advantage, under random circuit assumptions and depolarizing noise. However, their efficiency had not been demonstrated for more realistic nonunital noise models, such as amplitude damping, which better capture the behaviour of current hardware. We close this gap by adapting Pauli backpropagation to most single‑qubit noise channels, including nonunital noise, and proving that it remains efficient under these conditions. Our algorithm is parallelizable, memory‑efficient, and requires no assumptions on the locality of the observable, the geometry of the hardware, or the depth of the circuit. While hardware noise can strongly affect computational power in highly structured, fault‑tolerant circuits, our findings indicate little difference for most near‑term variational architectures. This establishes Pauli backpropagation as an efficient method for simulating noisy quantum devices under realistic noise assumptions.While these advances improve our ability to reproduce noisy expectation values of observables, many practical applications, including combinatorial optimization, require direct access to the bitstrings output by quantum circuits, as these encode candidate solutions. Classical replication of such noisy samples has so far been limited beyond the random circuit setting. To bridge this gap, we first focus on problems whose objective functions depend only on two‑body correlations, such as Max‑Cut, and apply Gaussian randomized rounding to the circuit's two‑qubit marginals. This produces classical samples whose expected cost is provably close to that of the noisy quantum device. Large‑scale simulations and experiments on IBMQ hardware confirm that the rounded samples reproduce the full energy distribution, with similar behaviour under various noise models. Coupled with Pauli backpropagation for efficiently estimating noisy marginals, this yields an end‑to‑end classical sampler operating in polynomial time for a broad class of variational circuits. Our results therefore show that sufficiently noisy quantum circuits for combinatorial optimization can be sampled from classically, clarifying where near‑term hardware may or may not outperform classical algorithms.Finally , we go beyond one and two‑body marginals and present an algorithm that samples valid bitstrings directly from a maximum‑entropy distribution matching the given expectation values of the quantum circuit. This method does not rely on the locality or sparsity of the problem's Hamiltonian and performs well across a variety of quantum circuits, including deep circuits on many qubits. When combined with Pauli propagation or error mitigation, it provides a practical pipeline for obtaining samples from quantum circuits in the near term, where full error correction might not yet be available.
We introduce Quantum-Informed Surrogate Sampling (QISS), a post-processing framework that generates candidate solutions to combinatorial optimization problems from low-weight correlations of shallow quantum circuits. The quantum device estimates local observables, which are directly accessible by repeated measurements and for which a wide range of error-mitigation tools are available, while candidate solutions are generated classically without explicit dependence on the combinatorial optimization problem itself. We evaluate QISS on Maximum Cut and Maximum Independent Set problems on N variables and show that only O(N) low-order correlators from shallow circuits suffice to produce competitive solutions that surpass vanilla QAOA. For MaxCut on 3-regular graphs, QISS from p=3 QAOA correlators outperforms vanilla QAOA at p=17 on average, with further improvements possible by warm-starting QAOA. We validate the procedure on the 54-qubit IQM Emerald quantum device and demonstrate its noise resilience. Our results support a regime for near-term optimization in which shallow circuits serve not as direct samplers but as generators of informative statistics for scalable classical sampling.
Exploiting the coherence and entanglement of quantum many-qubit states, quantum computing can significantly surpass classical algorithms, making it possible to factor large numbers, solve linear equations, simulate many-body quantum systems, etc., in a reasonable time. With the rapid development of quantum computing hardware, many attention has been drawn to explore how quantum computers could go beyond the limit of classical computation. Owing to the need of a universal fault-tolerant quantum computer for many existing quantum algorithms, such as Shor’s factoring algorithm, and considering the limit of near-term quantum devices with small qubit numbers and short coherence times, many recent works focused on the exploration of demonstrating quantum advantages using noisy intermediate-scaled quantum devices and shallow circuits, and hence some sampling problems have been proposed as the candidates for quantum advantage demonstration. This review summarizes quantum advantage problems that are realizable on current quantum hardware. We focus on two notable problems—random circuit simulation and boson sampling—and consider recent theoretical and experimental progresses. After the respective demonstrations of these two types of quantum advantages on superconducting and optical quantum platforms, we expect current and near-term quantum devices could be employed for demonstrating quantum advantages in general problems.
Quantum computing has emerged as a powerful computational paradigm capable of solving problems beyond the reach of classical computers. However, today’s quantum computers are noisy, posing challenges to obtaining accurate results. Here, we explore the impact of noise on quantum computing, focusing on the challenges in sampling bit strings from noisy quantum computers and the implications for optimization and machine learning. We formally quantify the sampling overhead to extract good samples from noisy quantum computers and relate it to the layer fidelity, a metric to determine the performance of noisy quantum processors. Further, we show how this allows us to use the conditional value at risk of noisy samples to determine provable bounds on noise-free expectation values. We discuss how to leverage these bounds for different algorithms and demonstrate our findings through experiments on real quantum computers involving up to 127 qubits. The results show strong alignment with theoretical predictions. In this study, the authors investigate the impact of noise on quantum computing with a focus on the challenges in sampling bit strings from noisy quantum computers, which has implications for optimization and machine learning.
We consider quantum circuits consisting of d layers of nearest-neighbor two-qubit gates acting on n qubits arranged on a line, where every qubit is independently depolarized with a constant probability before each layer. We describe a randomized parallel algorithm which samples from the output distribution of any such circuit to within total variation error δ, with parallel runtime 2O(d)loglog(n/δ) and n2O(d) elementary real-arithmetic operations. Without noise, the same approach gives an exact sampler with parallel runtime O(logn). For constant depth, we further show that the input/output behavior of the noisy quantum circuit is reproduced up to a constant error by a randomized AC0-circuit, that is, a Boolean circuit of polynomial size and constant depth with unbounded fan-in AND and OR gates and NOT gates. Consequently, every relation problem solved by a noisy constant-depth quantum circuit in one dimension is also solved, with essentially the same success probability, by a randomized AC0-circuit. This rules out, for noisy circuits in one dimension, the unconditional quantum advantage established for noisy shallow circuits in two and three dimensions, where polynomial-size classical circuits over the same gate set require depth Ω(logn/loglogn). Our algorithm exploits the fact that depolarizing noise cuts a one-dimensional circuit into independent pieces of logarithmic width, each of which can be sampled exactly in parallel.
Abstract Demonstrating quantum advantage on noisy quantum devices is one of the most important tasks of quantum computing research in the near future. Random quantum circuit sampling is proposed as the most promising approach to the task and has been implemented in experiments for achieving quantum advantage. A series of encouraging computational complexity results, based on some plausible assumptions, show that this task is impossible to complete efficiently by classical computers. However, in practical experiments on noisy quantum devices, the approximate average‐case hardness and the effects of the noise need to be further checked. The competition between the classical simulation algorithm (mainly based on tensor network algorithms) and noisy quantum devices is the explicit way to understand quantum advantage in practice. This review briefly overviews the computational complexity arguments for the hardness of classical simulation of random quantum circuits sampling, then focus on various methods of classical simulation of quantum circuits based on tensor network method.