Noise detrimentally affects quantum computations so that they not only become less accurate but also easier to simulate classically as systems scale up. We construct a classical simulation algorithm, lowesa (low weight efficient simulation algorithm), for estimating expectation values of noisy parameterised quantum circuits with a fixed observable. It combines previous results on spectral analysis of parameterised circuits with Pauli back-propagation and recent ideas for simulations of noisy random circuits. We show, under some conditions on the circuits and mild assumptions on noise, that lowesa gives an efficient, polynomial algorithm in the number of qubits (and depth), with approximation error that vanishes exponentially in the physical error rate and a controllable cutoff parameter. This is valid for any expectation value that may be efficiently evaluated on a quantum computer. We discuss the practical limitations of the method for circuit classes with correlated parameters and its scaling with decreasing error rates.
Noise detrimentally affects quantum computations so that they not only become less accurate but also easier to simulate classically as systems scale up. We construct a classical simulation algorithm, lowesa (low weight efficient simulation algorithm), for estimating expectation values of noisy parameterised quantum circuits with a fixed observable. It combines previous results on spectral analysis of parameterised circuits with Pauli back-propagation and recent ideas for simulations of noisy random circuits. We show, under some conditions on the circuits and mild assumptions on noise, that lowesa gives an efficient, polynomial algorithm in the number of qubits (and depth), with approximation error that vanishes exponentially in the physical error rate and a controllable cutoff parameter. This is valid for any expectation value that may be efficiently evaluated on a quantum computer. We discuss the practical limitations of the method for circuit classes with correlated parameters and its scaling with decreasing error rates.
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 present a polynomial-time classical algorithm for estimating expectation values of arbitrary observables on typical quantum circuits under any incoherent local noise, including non-unital or dephasing. Although previous research demonstrated that some carefully designed quantum circuits affected by non-unital noise cannot be efficiently simulated, we show that this does not apply to average-case circuits, as these can be efficiently simulated using Pauli-path methods. Specifically, we prove that, with high probability over the circuit gates’ choice, Pauli propagation algorithms with tailored truncation strategies achieve an inversely polynomially small simulation error. This result holds for arbitrary circuit topologies and for any local noise, under the assumption that the distribution of each circuit layer is invariant under single-qubit random gates. Under the same minimal assumptions, we also prove that most noisy circuits can be truncated to an effective logarithmic depth for the task of estimating expectation values of observables, thus generalizing prior results to a significantly broader class of circuit ensembles. We further numerically validate our algorithm with simulations on a 6 × 6 lattice of qubits under the effects of amplitude damping and dephasing noise, as well as real-time dynamics on an 11 × 11 lattice of qubits affected by amplitude damping.
Large-scale variational quantum algorithms are widely recognized as a potential pathway to achieve practical quantum advantages. However, the presence of quantum noise might suppress and undermine these advantages, which blurs the boundaries of classical simulability. To gain further clarity on this matter, we present a novel polynomial-scale method based on the path integral of observable's backpropagation on Pauli paths (OBPPP). This method efficiently approximates expectation values of operators in variational quantum algorithms with bounded truncation error in the presence of single-qubit Pauli noise. Theoretically, we rigorously prove: (i) For a constant minimal nonzero noise rate γ, OBPPP's time and space complexity exhibit a polynomial relationship with the number of qubits n, the circuit depth L. (ii) For variable γ, in scenarios where more than two nonzero noise factors exist, the complexity remains Poly(n,L) if γ exceeds 1/logL, but grows exponential with L when γ falls below 1/L. Numerically, we conduct classical simulations of IBM's zero-noise extrapolated experimental results on the 127-qubit Eagle processor [Y. Kim et al., Evidence for the utility of quantum computing before fault tolerance, Nature (London) 618, 500 (2023).NATUAS0028-083610.1038/s41586-023-06096-3]. Our method attains higher accuracy and faster runtime compared to the quantum device. Furthermore, our approach allows us to simulate noisy outcomes, enabling accurate reproduction of IBM's unmitigated results that directly correspond to raw experimental observations. Our research reveals the vital role of noise in classical simulations and the derived method is general in computing the expected value for a broad class of quantum circuits and can be applied in the verification of quantum computers.
We provide a polynomial-time classical algorithm for noisy quantum circuits. The algorithm computes the expectation value of any observable for any circuit, with a small average error over input states drawn from an ensemble (e.g., the computational basis). Our approach is based upon the intuition that noise exponentially damps nonlocal correlations relative to local correlations. This enables one to classically simulate a noisy quantum circuit by keeping track of only the dynamics of local quantum information. Our algorithm also enables sampling from the output distribution of a circuit in quasipolynomial time, so long as the distribution anticoncentrates. A number of implications are discussed, including a fundamental limit on the efficacy of noise mitigation strategies: For constant noise rates, any quantum circuit for which error mitigation succeeds in polynomial-time on most input states can also be classically simulated in polynomial-time on most input states. Our algorithms scale exponentially in the inverse noise rate, which is fundamental and makes them impractical for current quantum devices.
Magic states are essential for universal quantum computation and are widely viewed as a key source of quantum advantage, yet in realistic devices they are inevitably noisy. In this work, we characterize how noise on injected magic resources changes the classical simulability of quantum circuits and when it induces a transition from classically intractable behavior to efficient classical simulation. We adopt a resource-centric noise model in which only the injected magic components are noisy, while the baseline states, operations, and measurements belong to an efficiently simulable family. Within this setting, we develop an approximate classical sampling algorithm with controlled error and prove explicit noise-dependent conditions under which the algorithm runs in polynomial time. Our framework applies to both qubit circuits with Clifford baselines and fermionic circuits with matchgate baselines, covering representative noise channels such as dephasing and particle loss. We complement the analysis with numerical estimates of the simulation cost, providing concrete thresholds and runtime scaling across practically relevant parameter regimes.
One of the crucial steps in building a scalable quantum computer is to identify the noise sources which lead to errors in the process of quantum evolution. Different implementations come with multiple hardware-dependent sources of noise and decoherence making the problem of their detection manyfoldly more complex. We develop a randomized benchmarking algorithm which uses Weyl unitaries to efficiently identify and learn a mixture of error models which occur during the computation. We provide an efficiently computable estimate of the overhead required to compute expectation values on outputs of the noisy circuit relying only on the locality of the interactions and no further assumptions on the circuit structure. The overhead decreases with the noise rate and this enables us to compute analytic noise bounds that imply efficient classical simulability. We apply our methods to ansatz circuits that appear in the variational quantum eigensolver and establish an upper bound on classical simulation complexity as a function of noise, identifying regimes when they become classically efficiently simulatable.