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arXiv:1902.02258 · Physical Review A

Noise in boson sampling and the threshold of efficient classical simulatability

V. S. Shchesnovich

Published Jul 24, 2019• 31 citations
arXivHTMLPDF

Abstract

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

DOI
10.1103/physreva.100.012340

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To simulate noisy boson sampling approximating it by only the lower-order multi-boson interferences (e.g., by a smaller number of interfering bosons and classical particles) is very popular idea. I show that the output data from any such classical simulations can be efficiently distinguished from that of the quantum device they try to simulate, even with finite noise in the latter. The distinguishing datasets can be the experimental estimates of some large probabilities, a wide class of such is presented. This is a sequel of \textit{Quantum} \textbf{5}, 423 (2021), where I present more accessible account of the main result enhanced by additional insight on the contribution from the higher-order multi-boson interferences in presence of noise.

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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 \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ, i.e., \ensuremathϵ=\ensuremathΩ(1)\ensuremath{\epsilon}=\mathrm{\ensuremath{\Omega}}(1)\ensuremathϵ=\ensuremathΩ(1) as N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞. 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., \ensuremathϵ=O(1/N)\ensuremath{\epsilon}=O(1/N)\ensuremathϵ=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→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞ for arbitrarily small noise amplitude, i.e., \ensuremathϵ=o(1)\ensuremath{\epsilon}=o(1)\ensuremathϵ=o(1) as N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞; 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
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.
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.
Boson sampling, a computational problem conjectured to be hard to simulate on a classical machine, is a promising candidate for an experimental demonstration of quantum advantage using bosons. However, inevitable experimental noise and imperfections, such as loss in the interferometer and random counts at the detectors, could challenge the sampling task from entering the regime where quantum advantage is achievable. In this work we introduce higher-order nonlinearities as a means to enhance the computational complexity of the problem and the protocol's robustness against noise, i.e., to increase the noise threshold that allows us to perform an efficient classical simulation of the problem. Using a phase-space method based on the negativity volume of the relevant quasiprobability distributions, we establish a necessary nonclassicality condition that any experimental proof of quantum advantage must satisfy. Our results indicate that the addition of single-mode Kerr nonlinearity at the input-state preparation level, while retaining a linear-optical evolution, makes the boson-sampling protocol more robust against noise and consequently relaxes the constraints on the noise parameters required to show quantum advantage.
Abstract It is important to know noise levels of boson sampling in order to cautiously demonstrate the quantum computational advantage or realize certain tasks. Based on those statistical benchmark methods such as the correlators and clouds, which are initially proposed to discriminate boson sampling and other mockups, we quantificationally evaluate noises of photon partial distinguishability and photon loss compensated by dark counts. This is feasible owing to the fact that the output distribution unbalances are suppressed by noises, which are actually results of multi-photon interference. This is why the evaluation performance is better when high order correlators or corresponding clouds are employed. Our results indicate that the statistical benchmark methods can also work in the task of evaluating noises of boson sampling. An effective scheme is also introduced to fast simulate noisy samples, especially those with photon partial distinguishability.
Boson sampling stands out as a promising approach toward experimental demonstration of quantum computational advantage. However, the presence of physical noise in near-term experiments hinders the realization of quantum computational advantage with boson sampling. Since physical noise in near-term boson-sampling devices is inevitable, precise characterization of the boundary of noise rates where the classical intractability of boson sampling is maintained is crucial for quantum computational advantage using near-term devices. In this work, we identify the level of partial-distinguishability noise that upholds the classical intractability of boson sampling. We find that boson sampling with on average O ( log  N ) distinguishable photons out of N input photons maintains the equivalent complexity to the ideal boson-sampling case. By providing strong complexity-theoretical evidence for the classical intractability of noisy boson sampling, we expect that our findings will ultimately facilitate the demonstration of quantum computational advantage with noisy boson-sampling experiments in the near future.
To simulate noisy boson sampling approximating it by only the lower-order multi-boson interferences (e.g., by a smaller number of interfering bosons and classical particles) is very popular idea. I show that the output data from any such classical simulations can be efficiently distinguished from that of the quantum device they try to simulate, even with finite noise in the latter. The distinguishing datasets can be the experimental estimates of some large probabilities, a wide class of such is presented. This is a sequel of \textit{Quantum} \textbf{5}, 423 (2021), where I present more accessible account of the main result enhanced by additional insight on the contribution from the higher-order multi-boson interferences in presence of noise.