Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2510.24218 · Quantum

Pilot-Wave Simulator: Exact Classical Sampling from Ideal and Noisy Quantum Circuits up to Hundreds of Qubits

Gleb Vyacheslavovich Kalachev, P. A. Mosharev, Zuoheng Zou, Pavel Panteleev, Man‐Hong Yung

Published Jul 23, 2026• 0 citations
arXivHTMLPDF

Abstract

Quantum circuit simulators running on classical computers offer a vital platform for designing, testing, and optimizing quantum algorithms, driving innovation despite limited access to real quantum hardware. However, their scalability is inherently constrained by exponential memory and computational overhead, which restricts accurate simulation of large-scale quantum circuits and often results in approximate output distributions. Here, we propose an exact sampling algorithm that integrates tensor network contraction techniques with a Markov process, wherein a classical state evolves according to the local structure of the quantum circuit. As a demonstration, we target the challenge of generating samples from ideal and noisy QAOA circuits with up to 476 qubits, incorporating both depolarizing and amplitude damping noise models. These results enable further validation of several assumptions and conjectures at a scale previously out of reach, significantly expanding the scope of classical simulation in quantum algorithm research.

DOI
10.22331/q-2026-07-23-2173

Related papers

Classical simulation of noisy quantum circuits via locally entanglement-optimal unravelings

Simon Cichy, Paul K. Faehrmann, Lennart Bittel +2 more

Classical simulations of noisy quantum circuits are instrumental to our understanding of the behavior of real-world quantum systems and the identification of regimes where one expects quantum advantage. In this work, we present a highly parallelizable tensor-network-based classical algorithm, equipped with rigorous accuracy guarantees, for simulating n -qubit quantum circuits with arbitrary single-qubit noise. Our algorithm represents the state of a noisy quantum system by a particular ensemble of matrix product states from which we stochastically sample. Each pure state evolved under a single-qubit noise process is then represented by the ensemble of states that achieves the minimal average entanglement (the entanglement of formation) between the noisy qubit and the remainder. This approach lets us use a more compact representation of the quantum state for a given accuracy requirement and noise level. For a given maximum bond dimension χ and circuit, our algorithm comes with an upper bound on the simulation error, runs in poly ( n , χ ) -time, and improves upon related prior work (1) in scope by extending analytic methods from the three commonly considered noise models to general single-qubit noise, (2) in performance by deriving an exact, closed-form solution to the local entanglement minimization problem, previously approached via variational heuristics, thereby guaranteeing local optimality without numerical optimization overhead, and (3) in conceptual contribution by showing that the fixed unraveling used in prior work becomes equivalent to our choice of unraveling in the special case of depolarizing, dephasing, and amplitude damping noise acting on a maximally entangled state.

arXiv:2508.05745•Physical Review A•Jul 9, 2026•0 citations

Approximation Algorithm for Noisy Quantum Circuit Simulation

Mingyu Huang, Ji Guan, Wang Sheng Fang +1 more

Simulating noisy quantum circuits is vital in de-signing and verifying quantum algorithms in the current NISQ (Noisy Intermediate-Scale Quantum) era, where quantum noise is unavoidable. However, it is much more inefficient than the classical counterpart because of the quantum state explosion problem (the dimension of state space is exponential in the number of qubits) and the complex (non-unitary) representation of noises. Consequently, only noisy circuits with up to about 50 qubits can be simulated approximately well. To improve the scalability of the circuits that can be simulated, this paper introduces a novel approximation algorithm for simulating noisy quantum circuits when the noisy effectiveness is insignificant. The algorithm is based on a new tensor network diagram for the noisy simulation and uses the singular value decomposition to approximate the tensors of quantum noises in the diagram. The contraction of the tensor network diagram is implemented on Google's TensorNetwork. The effectiveness and utility of the algorithm are demonstrated by experimenting on a series of practical quantum circuits with realistic superconducting noise models. As a result, our algorithm can approximately simulate quantum circuits with up to 225 qubits and 20 noises (within about 1.8 hours). In particular, our method offers a speedup over the commonly-used approximation (sampling) algorithm - quantum trajectories method [1]. Furthermore, our approach can significantly reduce the number of samples in the quantum trajectories method when the noise rate is small enough,

arXiv:2211.17028•Design, Automation, and Test in Europe•Mar 25, 2024•1 citation

Comparative Study of Sampling-Based Simulation Costs of Noisy Quantum Circuits

Shigeo Hakkaku, Keisuke Fujii

Noise in quantum operations often negates the advantage of quantum computation. However, most classical simulations of quantum computers calculate the ideal probability amplitudes by either storing full state vectors or using sophisticated tensor-network contractions. Here we investigate sampling-based classical simulation methods for noisy quantum circuits. Specifically, we characterize the simulation costs of two major schemes, stabilizer-state sampling of magic states and Heisenberg propagation, for quantum circuits subject to stochastic Pauli noise, such as depolarizing and dephasing noise. To this end, we introduce several techniques for the stabilizer-state sampling to reduce the simulation costs under such noise. It is revealed that in the low-noise regime, stabilizer-state sampling results in a smaller sampling cost, while Heisenberg propagation is better in the high-noise regime. Furthermore, for a high depolarizing noise rate of approximately 10%, these methods provide better scaling than that given by the low-rank stabilizer decomposition. We believe that this knowledge of classical simulation costs is useful to achieve possible quantum advantage on near-term noisy quantum devices as well as efficient classical simulation methods.

arXiv:2011.06233•Physical Review Applied•Jun 10, 2021•4 citations

Noisy quantum circuit simulation with the tensor jump method

Maximilian Fröhlich, Aaron Sander, Martin Eigel +2 more

Classical simulation of noisy quantum circuits is essential for validating algorithms, benchmarking hardware, and assessing error-mitigation strategies, but remains limited by the exponential cost of density-matrix methods and the high variance of standard trajectory sampling. We introduce a variance-aware tensor network framework that combines the tensor jump method with local TDVP gate evolution on matrix product states and sparse Pauli-Lindblad hardware noise models. Gates are applied as short variational evolutions on the MPS manifold, while noise is sampled per circuit window from Pauli-Lindblad jump sets with state-independent hazards and dissipative contractions that reduce to irrelevant global factors after renormalization. The method supports correlated multi-qubit Lindblad noise consistent with hardware connectivity, including long-range operators on non-adjacent qubits, enabling direct simulation of crosstalk and other connectivity-induced errors beyond local noise models. We develop two unbiased variance-aware unravelings. An analog unitary-mixture unraveling matches the Lindblad generator exactly under symmetric Gaussian or two-point angle laws, while a projector-jump unraveling yields state-independent hazards and closed-form variance laws. Both retain the standard 1/sqrt(N) Monte Carlo convergence but with reduced prefactors. Empirically, projector sampling strongly reduces trajectory variance and bond-dimension growth across many circuit architectures, whereas analog sampling is most effective at weak noise. We demonstrate accurate, scalable noisy-circuit simulation on a 25-qubit noisy XY quench and IBM's 127-qubit kicked-Ising benchmark with long-range depolarizing noise, achieving reduced Monte Carlo variance and favorable MPS bond-dimension growth compared with standard Kraus-insertion baselines.

quant-pharXiv:2607.01323•Jul 3, 2026•0 citations

A polynomial-time classical sampler for noisy quantum circuits from statistical mechanics

Jon Nelson, Joel Rajakumar, Chao Yin +2 more

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 $Θ(p^{-1} \log p^{-1})$ 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.

quant-pharXiv:2610.00548•Oct 2, 2026•0 citations

Logical abstractions for noisy variational Quantum algorithm simulation

Yipeng Huang, Steven Holtzen, Todd D. Millstein +2 more

Due to the unreliability and limited capacity of existing quantum computer prototypes, quantum circuit simulation continues to be a vital tool for validating next generation quantum computers and for studying variational quantum algorithms, which are among the leading candidates for useful quantum computation. Existing quantum circuit simulators do not address the common traits of variational algorithms, namely: 1) their ability to work with noisy qubits and operations, 2) their repeated execution of the same circuits but with different parameters, and 3) the fact that they sample from circuit final wavefunctions to drive a classical optimization routine. We present a quantum circuit simulation toolchain based on logical abstractions targeted for simulating variational algorithms. Our proposed toolchain encodes quantum amplitudes and noise probabilities in a probabilistic graphical model, and it compiles the circuits to logical formulas that support efficient repeated simulation of and sampling from quantum circuits for different parameters. Compared to state-of-the-art state vector and density matrix quantum circuit simulators, our simulation approach offers greater performance when sampling from noisy circuits with at least eight to 20 qubits and with around 12 operations on each qubit, making the approach ideal for simulating near-term variational quantum algorithms. And for simulating noise-free shallow quantum circuits with 32 qubits, our simulation approach offers a 66× reduction in sampling cost versus quantum circuit simulation techniques based on tensor network contraction.

arXiv:2103.17226•International Conference on Architectural Support for Programming Languages and Operating Systems (ASPLOS)•Apr 11, 2021•16 citations
Quantum circuit simulators running on classical computers offer a vital platform for designing, testing, and optimizing quantum algorithms, driving innovation despite limited access to real quantum hardware. However, their scalability is inherently constrained by exponential memory and computational overhead, which restricts accurate simulation of large-scale quantum circuits and often results in approximate output distributions. Here, we propose an exact sampling algorithm that integrates tensor network contraction techniques with a Markov process, wherein a classical state evolves according to the local structure of the quantum circuit. As a demonstration, we target the challenge of generating samples from ideal and noisy QAOA circuits with up to 476 qubits, incorporating both depolarizing and amplitude damping noise models. These results enable further validation of several assumptions and conjectures at a scale previously out of reach, significantly expanding the scope of classical simulation in quantum algorithm research.
Classical simulations of noisy quantum circuits are instrumental to our understanding of the behavior of real-world quantum systems and the identification of regimes where one expects quantum advantage. In this work, we present a highly parallelizable tensor-network-based classical algorithm, equipped with rigorous accuracy guarantees, for simulating n -qubit quantum circuits with arbitrary single-qubit noise. Our algorithm represents the state of a noisy quantum system by a particular ensemble of matrix product states from which we stochastically sample. Each pure state evolved under a single-qubit noise process is then represented by the ensemble of states that achieves the minimal average entanglement (the entanglement of formation) between the noisy qubit and the remainder. This approach lets us use a more compact representation of the quantum state for a given accuracy requirement and noise level. For a given maximum bond dimension χ and circuit, our algorithm comes with an upper bound on the simulation error, runs in poly ( n , χ ) -time, and improves upon related prior work (1) in scope by extending analytic methods from the three commonly considered noise models to general single-qubit noise, (2) in performance by deriving an exact, closed-form solution to the local entanglement minimization problem, previously approached via variational heuristics, thereby guaranteeing local optimality without numerical optimization overhead, and (3) in conceptual contribution by showing that the fixed unraveling used in prior work becomes equivalent to our choice of unraveling in the special case of depolarizing, dephasing, and amplitude damping noise acting on a maximally entangled state.
Simulating noisy quantum circuits is vital in de-signing and verifying quantum algorithms in the current NISQ (Noisy Intermediate-Scale Quantum) era, where quantum noise is unavoidable. However, it is much more inefficient than the classical counterpart because of the quantum state explosion problem (the dimension of state space is exponential in the number of qubits) and the complex (non-unitary) representation of noises. Consequently, only noisy circuits with up to about 50 qubits can be simulated approximately well. To improve the scalability of the circuits that can be simulated, this paper introduces a novel approximation algorithm for simulating noisy quantum circuits when the noisy effectiveness is insignificant. The algorithm is based on a new tensor network diagram for the noisy simulation and uses the singular value decomposition to approximate the tensors of quantum noises in the diagram. The contraction of the tensor network diagram is implemented on Google's TensorNetwork. The effectiveness and utility of the algorithm are demonstrated by experimenting on a series of practical quantum circuits with realistic superconducting noise models. As a result, our algorithm can approximately simulate quantum circuits with up to 225 qubits and 20 noises (within about 1.8 hours). In particular, our method offers a speedup over the commonly-used approximation (sampling) algorithm - quantum trajectories method [1]. Furthermore, our approach can significantly reduce the number of samples in the quantum trajectories method when the noise rate is small enough,
Noise in quantum operations often negates the advantage of quantum computation. However, most classical simulations of quantum computers calculate the ideal probability amplitudes by either storing full state vectors or using sophisticated tensor-network contractions. Here we investigate sampling-based classical simulation methods for noisy quantum circuits. Specifically, we characterize the simulation costs of two major schemes, stabilizer-state sampling of magic states and Heisenberg propagation, for quantum circuits subject to stochastic Pauli noise, such as depolarizing and dephasing noise. To this end, we introduce several techniques for the stabilizer-state sampling to reduce the simulation costs under such noise. It is revealed that in the low-noise regime, stabilizer-state sampling results in a smaller sampling cost, while Heisenberg propagation is better in the high-noise regime. Furthermore, for a high depolarizing noise rate of approximately 10%, these methods provide better scaling than that given by the low-rank stabilizer decomposition. We believe that this knowledge of classical simulation costs is useful to achieve possible quantum advantage on near-term noisy quantum devices as well as efficient classical simulation methods.
Classical simulation of noisy quantum circuits is essential for validating algorithms, benchmarking hardware, and assessing error-mitigation strategies, but remains limited by the exponential cost of density-matrix methods and the high variance of standard trajectory sampling. We introduce a variance-aware tensor network framework that combines the tensor jump method with local TDVP gate evolution on matrix product states and sparse Pauli-Lindblad hardware noise models. Gates are applied as short variational evolutions on the MPS manifold, while noise is sampled per circuit window from Pauli-Lindblad jump sets with state-independent hazards and dissipative contractions that reduce to irrelevant global factors after renormalization. The method supports correlated multi-qubit Lindblad noise consistent with hardware connectivity, including long-range operators on non-adjacent qubits, enabling direct simulation of crosstalk and other connectivity-induced errors beyond local noise models. We develop two unbiased variance-aware unravelings. An analog unitary-mixture unraveling matches the Lindblad generator exactly under symmetric Gaussian or two-point angle laws, while a projector-jump unraveling yields state-independent hazards and closed-form variance laws. Both retain the standard 1/sqrt(N) Monte Carlo convergence but with reduced prefactors. Empirically, projector sampling strongly reduces trajectory variance and bond-dimension growth across many circuit architectures, whereas analog sampling is most effective at weak noise. We demonstrate accurate, scalable noisy-circuit simulation on a 25-qubit noisy XY quench and IBM's 127-qubit kicked-Ising benchmark with long-range depolarizing noise, achieving reduced Monte Carlo variance and favorable MPS bond-dimension growth compared with standard Kraus-insertion baselines.
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 ppp after Θ(p−1log⁡p−1)Θ(p^{-1} \log p^{-1})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.
Due to the unreliability and limited capacity of existing quantum computer prototypes, quantum circuit simulation continues to be a vital tool for validating next generation quantum computers and for studying variational quantum algorithms, which are among the leading candidates for useful quantum computation. Existing quantum circuit simulators do not address the common traits of variational algorithms, namely: 1) their ability to work with noisy qubits and operations, 2) their repeated execution of the same circuits but with different parameters, and 3) the fact that they sample from circuit final wavefunctions to drive a classical optimization routine. We present a quantum circuit simulation toolchain based on logical abstractions targeted for simulating variational algorithms. Our proposed toolchain encodes quantum amplitudes and noise probabilities in a probabilistic graphical model, and it compiles the circuits to logical formulas that support efficient repeated simulation of and sampling from quantum circuits for different parameters. Compared to state-of-the-art state vector and density matrix quantum circuit simulators, our simulation approach offers greater performance when sampling from noisy circuits with at least eight to 20 qubits and with around 12 operations on each qubit, making the approach ideal for simulating near-term variational quantum algorithms. And for simulating noise-free shallow quantum circuits with 32 qubits, our simulation approach offers a 66× reduction in sampling cost versus quantum circuit simulation techniques based on tensor network contraction.