As quantum devices continue to grow in size but remain affected by noise, it is crucial to determine when and how they can outperform classical computers on practical tasks. A central piece in this effort is to develop the most efficient classical simulation algorithms possible. Among the most promising approaches are Pauli backpropagation algorithms, which have already demonstrated their ability to efficiently simulate certain classes of parameterized quantum circuits-a leading contender for near-term quantum advantage-under random circuit assumptions and depolarizing noise. However, their efficiency was not previously established for more realistic nonunital noise models, such as amplitude damping, that better capture noise on existing hardware. Here, we close this gap by adapting Pauli backpropagation to nonunital noise, proving that it remains efficient even under these more challenging conditions. Our proof leverages a refined combinatorial analysis to handle the complexities introduced by nonunital channels, thus strengthening Pauli backpropagation as a powerful tool for simulating near-term quantum devices.
As quantum devices continue to grow in size but remain affected by noise, it is crucial to determine when and how they can outperform classical computers on practical tasks. A central piece in this effort is to develop the most efficient classical simulation algorithms possible. Among the most promising approaches are Pauli backpropagation algorithms, which have already demonstrated their ability to efficiently simulate certain classes of parameterized quantum circuits-a leading contender for near-term quantum advantage-under random circuit assumptions and depolarizing noise. However, their efficiency was not previously established for more realistic nonunital noise models, such as amplitude damping, that better capture noise on existing hardware. Here, we close this gap by adapting Pauli backpropagation to nonunital noise, proving that it remains efficient even under these more challenging conditions. Our proof leverages a refined combinatorial analysis to handle the complexities introduced by nonunital channels, thus strengthening Pauli backpropagation as a powerful tool for simulating near-term quantum devices.
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.
For random quantum circuits on n qubits of depth Θ ( log n ) with depolarizing noise, the task of sampling from the output state can be efficiently performed classically using a Pauli path method \cite{Aharonov2023_paulipaths} . This paper aims to study the performance of this method beyond random circuits. We first consider the classical simulation of local observables in circuits composed of Clifford and T gates – going beyond the average case analysis, we derive sufficient conditions for simulatability in terms of the noise rate and the fraction of gates that are T gates, and show that if noise is introduced at a faster rate than T gates, the simulation becomes classically easy. As an application of this result, we study 2D QAOA circuits that attempt to find low-energy states of classical Ising models on general graphs. There, our results shows that for hard instances of the problem, which correspond to Ising model's graph being geometrically non-local, a QAOA algorithm mapped to a geometrically local circuit architecture using SWAP gates does not have any asymptotic advantage over classical algorithms if depolarized at a constant rate. Finally, we illustrate instances where the Pauli path method fails to give the correct result, and also initiate a study of the trade-off between fragility to noise and classical complexity of simulating a given quantum circuit.
It is widely accepted that noisy quantum devices are limited to logarithmic depth circuits unless midcircuit measurements and error correction are employed. However, this conclusion holds only for unital error channels, such as depolarizing noise. Building on the idea of the “quantum refrigerator” Ben-Or [Quantum refrigerator, arXiv1301.1995 (2013)], we improve upon previous results and show that geometrically local circuits in the presence of nonunital noise, in any dimension d ≥ 1 , can correct errors without midcircuit measurements and extend computation to any depth, with only polylogarithmic overhead in the depth and the number of qubits. This implies that local quantum dynamics subjected to sufficiently weak nonunital noise are computationally universal and nearly as hard to simulate as noiseless dynamics. Additionally, we quantify the contraction property of local random circuits in the presence of nonunital noise.
Developing classical simulation algorithms for noisy quantum circuits is essential to delineating the limits of quantum advantage. Existing classical sampling approaches for general circuits require circuit depths to grow logarithmically with system size, so that noise drives the global output state close to a trivial state. Here we show that local accumulation of noise at a depth independent of system size is already sufficient. Specifically, we prove that any geometrically local circuit composed of unital operations and interspersed with single-qubit depolarizing noise of strength p after Θ(p−1logp−1) depth can be approximately sampled from a polynomial-time classical computer. This generalizes existing results that impose anticoncentration assumptions or non-universal gate sets in order to obtain a classical sampler at such shallow depths. Our proof maps the output state of the circuit to a polymer model in statistical mechanics, and combines a convergent cluster expansion with hypercontractivity of the depolarizing channel.
In this work, drawing inspiration from the type of noise present in real hardware, we study the output distribution of random quantum circuits under practical nonunital noise sources with constant noise rates. We show that even in the presence of unital sources such as the depolarizing channel, the distribution, under the combined noise channel, never resembles a maximally entropic distribution at any depth. To show this, we prove that the output distribution of such circuits never anticoncentrates—meaning that it is never too “flat”—regardless of the depth of the circuit. This is in stark contrast to the behavior of noiseless random quantum circuits or those with only unital noise, both of which anticoncentrate at sufficiently large depths. As a consequence, our results shows that the complexity of random-circuit sampling under realistic noise is still an open question, since anticoncentration is a critical property exploited by both state-of-the-art classical hardness and easiness results. Published by the American Physical Society 2024