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:2301.11532

On classical simulation algorithms for noisy Boson Sampling

Changhun Oh, Liang Jiang, Bill Fefferman

Published Jan 27, 2023• 3 citations
arXivHTMLPDF

Abstract

We present a classical algorithm that approximately samples from the output distribution of certain noisy Boson Sampling experiments. This algorithm is inspired by a recent result of Aharonov, Gao, Landau, Liu and Vazirani and makes use of an observation originally due to Kalai and Kindler that the output probability of Boson Sampling experiments with a Gaussian noise model can be approximated by sparse low-degree polynomials. This observation alone does not suffice for classical sampling, because its marginal probabilities might not be approximated by sparse low-degree polynomials, and furthermore, the approximated probabilities might be negative. We solve this problem by employing the first quantization representation to give an algorithm for computing the marginal probabilities of these experiments. We prove that when the overall noise rate is constant, the algorithm runs in time quasi-polynomial in the number of input photons $N$ and accuracy. When the overall noise rate scales as $1-x_1^γ$ for constant $x_1$ and $γ=Ω(\log N)$, the running time becomes polynomial. Furthermore, we study noisy Boson Sampling with practically relevant noise models such as partial distinguishability and photon loss. We show that the same technique does not immediately apply in these settings, leaving open the possibility of a scalable demonstration of noisy quantum advantage for these noise models in certain parameter regimes.

DOI
10.48550/arxiv.2301.11532

Related papers

Distinguishing noisy boson sampling from classical simulations

V. S. Shchesnovich

Giving a convincing experimental evidence of the quantum supremacy over classical simulations is a challenging goal. Noise is considered to be the main problem in such a demonstration, hence it is urgent to understand the effect of noise. Recently found classical algorithms can efficiently approximate, to any small error, the output of boson sampling with finite-amplitude noise. In this work it is shown analytically and confirmed by numerical simulations that one can efficiently distinguish the output distribution of such a noisy boson sampling from the approximations accounting for low-order quantum multiboson interferences, what includes the mentioned classical algorithms. The number of samples required to tell apart the quantum and classical output distributions is strongly affected by the previously unexplored parameter: density of bosons, i.e., the ratio of total number of interfering bosons to number of input ports of interferometer. Such critical dependence is strikingly reminiscent of the quantum-to-classical transition in systems of identical particles, which sets in when the system size scales up while density of particles vanishes.

arXiv:1905.11458•Quantum•Mar 29, 2021•19 citations

Regimes of Classical Simulability for Noisy Gaussian Boson Sampling

Haoyu Qi, Daniel J. Brod, Nicolás Quesada +1 more

As a promising candidate for exhibiting quantum computational supremacy, Gaussian boson sampling (GBS) is designed to exploit the ease of experimental preparation of Gaussian states. However, sufficiently large and inevitable experimental noise might render GBS classically simulable. In this work, we formalize this intuition by establishing a sufficient condition for approximate polynomial-time classical simulation of noisy GBS-in the form of an inequality between the input squeezing parameter, the overall transmission rate, and the quality of photon detectors. Our result serves as a nonclassicality test that must be passed by any quantum computational supremacy demonstration based on GBS. We show that, for most linear-optical architectures, where photon loss increases exponentially with the circuit depth, noisy GBS loses its quantum advantage in the asymptotic limit. Our results thus delineate intermediate-sized regimes where GBS devices might considerably outperform classical computers for modest noise levels. Finally, we find that increasing the amount of input squeezing is helpful to evade our classical simulation algorithm, which suggests a potential route to mitigate photon loss.

arXiv:1905.12075•Physical Review Letters•Mar 13, 2020•87 citations

Noise in boson sampling and the threshold of efficient classical simulatability

V. S. Shchesnovich

We study the quantum to classical transition in boson sampling by analyzing how $N$-boson interference is affected by inevitable noise in an experimental setup. We adopt the Gaussian noise model of Kalai and Kindler for boson sampling and show that it appears from some realistic experimental imperfections. We reveal a connection between noise in boson sampling and partial distinguishability of bosons, which allows us to prove efficient classical simulatability of noisy no-collision boson sampling with finite noise amplitude $\ensuremath{\epsilon}$, i.e., $\ensuremath{\epsilon}=\mathrm{\ensuremath{\Omega}}(1)$ as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. On the other hand, using an equivalent representation of network noise as losses of bosons compensated by random (dark) counts of detectors, it is proven that for noise amplitude inversely proportional to total number of bosons, i.e., $\ensuremath{\epsilon}=O(1/N)$, noisy no-collision boson sampling is as hard to simulate classically as in the noiseless case. Moreover, the ratio of ``noise clicks'' (lost bosons compensated by dark counts) to the total number of bosons $N$ vanishes as $N\ensuremath{\rightarrow}\ensuremath{\infty}$ for arbitrarily small noise amplitude, i.e., $\ensuremath{\epsilon}=o(1)$ as $N\ensuremath{\rightarrow}\ensuremath{\infty}$; hence, we conjecture that such a noisy boson sampling is also hard to simulate classically. The results significantly relax the sufficient condition on noise in network components, such as two-mode beam splitters, for classical hardness of experimental boson sampling

arXiv:1902.02258•Physical Review A•Jul 24, 2019•31 citations

Efficient classical algorithm for estimating linear statistics of Boson Sampling

Benoit Seron, Hugo Thomas, Eduardo Araujo +3 more

Boson Sampling is a prominent candidate for the demonstration of quantum computational advantage but it remains unclear whether a large-scale boson sampler can find useful computational applications. The challenge is that to use a boson sampler to estimate, for example, a physical observable, it is necessary to coarse grain the outcome distribution, owing to the exponential size of the outcome space and the anti-concentration properties of the Boson Sampling distribution. In this work, we analyse the complexity of a specific type of coarse-graining of Boson Sampling distributions based on linear functions of the output photon occupation numbers, which we refer to as linear statistics. We present an efficient classical algorithm for approximating linear statistics of boson samplers within additive error for different kinds of input states, such as Fock states and squeezed states. This allows us to unify in the same framework recent results on efficient quantum-inspired classical algorithms for simulating molecular vibronic spectra, or efficient classical approximations of coarse-grained distributions based on detector binning. Additionally, we show how our algorithm can be used to classically evaluate a proposed one-way function based on Boson Sampling, while other proposals for cryptographic applications escape our classical simulation techniques. We leave open the question of classical simulability of non-linear statistics and connect it to the problem of computing transition amplitudes of linear-optical circuits with one layer of interactions.

quant-pharXiv:2609.35447•Sep 29, 2026•0 citations

Classical sampling from noisy Boson Sampling and the negative probabilities

V. S. Shchesnovich

It is known that, by accounting for the multiboson interferences up to a finite order, the output distribution of noisy Boson Sampling, with distinguishability of bosons serving as noise, can be approximately sampled from in a time polynomial in the total number of bosons. The drawback of this approach is that the joint probabilities of completely distinguishable bosons, i.e., those that do not interfere at all, have to be computed also. In trying to restore the ability to sample from the distinguishable bosons with computation of only the single-boson probabilities, one faces the following issue: the quantum probability factors in a convex-sum expression, if truncated to a finite order of multiboson interference, have, on average, a finite amount of negativity in a random interferometer. The truncated distribution does become a proper one, while allowing for sampling from it in a polynomial time, only in a vanishing domain close to the completely distinguishable bosons. Nevertheless, the conclusion that the negativity issue is inherent to all efficient classical approximations to noisy Boson Sampling may be premature. I outline the direction for a whole new program, which seem to point to a solution. However its success depends on the asymptotic behavior of the symmetric group characters, which is not known.

arXiv:2307.05344•Jul 11, 2023•0 citations

Boson sampling is fragile to loss

Youngrong Lim

Boson sampling experiments aim to demonstrate quantum computational advantage with increasing numbers of interfering photons. A central question is whether their computational hardness survives under photon loss. We show that even an arbitrarily small nonzero uniform loss rate permits efficient classical sampling in the average case when the rate remains fixed as the system grows. We exploit how loss suppresses correlations involving many photons to approximate the full output distribution using low correlation orders. The resulting samplers run in time polynomial in system size and inverse total variation error, with the error averaged over Haar random interferometers. The results apply to Fock state boson sampling with single-photon states in distinct input modes and arbitrary additional vacuum inputs, and to Gaussian boson sampling with equal squeezing of bounded strength in every input mode. Classical sampling is efficient even when the mean detected photon number grows linearly with system size, showing that large photon counts alone do not ensure computational hardness.

•Oct 21, 2026•0 citations
We present a classical algorithm that approximately samples from the output distribution of certain noisy Boson Sampling experiments. This algorithm is inspired by a recent result of Aharonov, Gao, Landau, Liu and Vazirani and makes use of an observation originally due to Kalai and Kindler that the output probability of Boson Sampling experiments with a Gaussian noise model can be approximated by sparse low-degree polynomials. This observation alone does not suffice for classical sampling, because its marginal probabilities might not be approximated by sparse low-degree polynomials, and furthermore, the approximated probabilities might be negative. We solve this problem by employing the first quantization representation to give an algorithm for computing the marginal probabilities of these experiments. We prove that when the overall noise rate is constant, the algorithm runs in time quasi-polynomial in the number of input photons NNN and accuracy. When the overall noise rate scales as 1−x1γ1-x_1^γ1−x1γ​ for constant x1x_1x1​ and γ=Ω(log⁡N)γ=Ω(\log N)γ=Ω(logN), the running time becomes polynomial. Furthermore, we study noisy Boson Sampling with practically relevant noise models such as partial distinguishability and photon loss. We show that the same technique does not immediately apply in these settings, leaving open the possibility of a scalable demonstration of noisy quantum advantage for these noise models in certain parameter regimes.
Giving a convincing experimental evidence of the quantum supremacy over classical simulations is a challenging goal. Noise is considered to be the main problem in such a demonstration, hence it is urgent to understand the effect of noise. Recently found classical algorithms can efficiently approximate, to any small error, the output of boson sampling with finite-amplitude noise. In this work it is shown analytically and confirmed by numerical simulations that one can efficiently distinguish the output distribution of such a noisy boson sampling from the approximations accounting for low-order quantum multiboson interferences, what includes the mentioned classical algorithms. The number of samples required to tell apart the quantum and classical output distributions is strongly affected by the previously unexplored parameter: density of bosons, i.e., the ratio of total number of interfering bosons to number of input ports of interferometer. Such critical dependence is strikingly reminiscent of the quantum-to-classical transition in systems of identical particles, which sets in when the system size scales up while density of particles vanishes.
As a promising candidate for exhibiting quantum computational supremacy, Gaussian boson sampling (GBS) is designed to exploit the ease of experimental preparation of Gaussian states. However, sufficiently large and inevitable experimental noise might render GBS classically simulable. In this work, we formalize this intuition by establishing a sufficient condition for approximate polynomial-time classical simulation of noisy GBS-in the form of an inequality between the input squeezing parameter, the overall transmission rate, and the quality of photon detectors. Our result serves as a nonclassicality test that must be passed by any quantum computational supremacy demonstration based on GBS. We show that, for most linear-optical architectures, where photon loss increases exponentially with the circuit depth, noisy GBS loses its quantum advantage in the asymptotic limit. Our results thus delineate intermediate-sized regimes where GBS devices might considerably outperform classical computers for modest noise levels. Finally, we find that increasing the amount of input squeezing is helpful to evade our classical simulation algorithm, which suggests a potential route to mitigate photon loss.
We study the quantum to classical transition in boson sampling by analyzing how NNN-boson interference is affected by inevitable noise in an experimental setup. We adopt the Gaussian noise model of Kalai and Kindler for boson sampling and show that it appears from some realistic experimental imperfections. We reveal a connection between noise in boson sampling and partial distinguishability of bosons, which allows us to prove efficient classical simulatability of noisy no-collision boson sampling with finite noise amplitude $\ensuremath{\epsilon}$, i.e., $\ensuremath{\epsilon}=\mathrm{\ensuremath{\Omega}}(1)$ as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. On the other hand, using an equivalent representation of network noise as losses of bosons compensated by random (dark) counts of detectors, it is proven that for noise amplitude inversely proportional to total number of bosons, i.e., $\ensuremath{\epsilon}=O(1/N)$, noisy no-collision boson sampling is as hard to simulate classically as in the noiseless case. Moreover, the ratio of ``noise clicks'' (lost bosons compensated by dark counts) to the total number of bosons NNN vanishes as $N\ensuremath{\rightarrow}\ensuremath{\infty}$ for arbitrarily small noise amplitude, i.e., $\ensuremath{\epsilon}=o(1)$ as $N\ensuremath{\rightarrow}\ensuremath{\infty}$; hence, we conjecture that such a noisy boson sampling is also hard to simulate classically. The results significantly relax the sufficient condition on noise in network components, such as two-mode beam splitters, for classical hardness of experimental boson sampling
Boson Sampling is a prominent candidate for the demonstration of quantum computational advantage but it remains unclear whether a large-scale boson sampler can find useful computational applications. The challenge is that to use a boson sampler to estimate, for example, a physical observable, it is necessary to coarse grain the outcome distribution, owing to the exponential size of the outcome space and the anti-concentration properties of the Boson Sampling distribution. In this work, we analyse the complexity of a specific type of coarse-graining of Boson Sampling distributions based on linear functions of the output photon occupation numbers, which we refer to as linear statistics. We present an efficient classical algorithm for approximating linear statistics of boson samplers within additive error for different kinds of input states, such as Fock states and squeezed states. This allows us to unify in the same framework recent results on efficient quantum-inspired classical algorithms for simulating molecular vibronic spectra, or efficient classical approximations of coarse-grained distributions based on detector binning. Additionally, we show how our algorithm can be used to classically evaluate a proposed one-way function based on Boson Sampling, while other proposals for cryptographic applications escape our classical simulation techniques. We leave open the question of classical simulability of non-linear statistics and connect it to the problem of computing transition amplitudes of linear-optical circuits with one layer of interactions.
It is known that, by accounting for the multiboson interferences up to a finite order, the output distribution of noisy Boson Sampling, with distinguishability of bosons serving as noise, can be approximately sampled from in a time polynomial in the total number of bosons. The drawback of this approach is that the joint probabilities of completely distinguishable bosons, i.e., those that do not interfere at all, have to be computed also. In trying to restore the ability to sample from the distinguishable bosons with computation of only the single-boson probabilities, one faces the following issue: the quantum probability factors in a convex-sum expression, if truncated to a finite order of multiboson interference, have, on average, a finite amount of negativity in a random interferometer. The truncated distribution does become a proper one, while allowing for sampling from it in a polynomial time, only in a vanishing domain close to the completely distinguishable bosons. Nevertheless, the conclusion that the negativity issue is inherent to all efficient classical approximations to noisy Boson Sampling may be premature. I outline the direction for a whole new program, which seem to point to a solution. However its success depends on the asymptotic behavior of the symmetric group characters, which is not known.
Boson sampling experiments aim to demonstrate quantum computational advantage with increasing numbers of interfering photons. A central question is whether their computational hardness survives under photon loss. We show that even an arbitrarily small nonzero uniform loss rate permits efficient classical sampling in the average case when the rate remains fixed as the system grows. We exploit how loss suppresses correlations involving many photons to approximate the full output distribution using low correlation orders. The resulting samplers run in time polynomial in system size and inverse total variation error, with the error averaged over Haar random interferometers. The results apply to Fock state boson sampling with single-photon states in distinct input modes and arbitrary additional vacuum inputs, and to Gaussian boson sampling with equal squeezing of bounded strength in every input mode. Classical sampling is efficient even when the mean detected photon number grows linearly with system size, showing that large photon counts alone do not ensure computational hardness.