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arXiv:1904.05494 · Scientific Reports

Classical simulation of boson sampling with sparse output

Wojciech Roga, Masahiro Takeoka

Published Sep 7, 2020• 10 citations
arXivHTMLPDF

Abstract

Boson sampling can simulate physical problems for which classical simulations are inefficient. However, not all problems simulated by boson sampling are classically intractable. We show explicit classical methods of finding boson sampling distributions when they are known to be highly sparse. In the methods, we first determine a few distributions from restricted number of detectors and then recover the full one using compressive sensing techniques. In general, the latter step could be of high complexity. However, we show that this problem can be reduced to solving an Ising model which under certain conditions can be done in polynomial time. Various extensions are discussed including a version involving quantum annealing. Hence, our results impact the understanding of the class of classically calculable problems. We indicate that boson samplers may be advantageous in dealing with problems which are not highly sparse. Finally, we suggest a hybrid method for problems of intermediate sparsity.

DOI
10.1038/s41598-020-71892-0

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Boson sampling can simulate physical problems for which classical simulations are inefficient. However, not all problems simulated by boson sampling are classically intractable. We show explicit classical methods of finding boson sampling distributions when they are known to be highly sparse. In the methods, we first determine a few distributions from restricted number of detectors and then recover the full one using compressive sensing techniques. In general, the latter step could be of high complexity. However, we show that this problem can be reduced to solving an Ising model which under certain conditions can be done in polynomial time. Various extensions are discussed including a version involving quantum annealing. Hence, our results impact the understanding of the class of classically calculable problems. We indicate that boson samplers may be advantageous in dealing with problems which are not highly sparse. Finally, we suggest a hybrid method for problems of intermediate sparsity.
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.
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.
The original formulation of the boson sampling problem assumed that little or no photon collisions occur. However, modern experimental realizations rely on setups where collisions are quite common, i.e., the number of photons M injected into the circuit is close to the number of detectors N. Here we present a classical algorithm that simulates a bosonic sampler: it calculates the probability of a given photon distribution at the interferometer outputs for a given distribution at the inputs. This algorithm is most effective in cases with multiple photon collisions, and in those cases, it outperforms known algorithms.
Quantum computers solve intractable problems which classically require an exponentially long time to compute. With the development of large-scale experiments that claim quantum advantage, a vital issue has now emerged. What are the errors, and how do they affect the complexity of the problem solved? Large-scale Gaussian boson sampling (GBS) experiments give an example in which random numbers are generated. Despite classical hardness, these have computable benchmarks for checking data validity. While there are other quantum computing architectures, Gaussian boson sampling is uniquely testable at all scales. Several large, pioneering quantum computing (QC) experiments have been carried out to investigate quantum advantage. Here, we introduce a highly scalable but classical algorithm that can solve GBS approximately. Our numerical simulation of the output count data is closer to the exact solution than current experiments up to 1152 modes. This algorithm outperforms all previous classical, approximate algorithms and scales efficiently to larger experiments. Our results show that effects beyond losses can cause the errors that allow classical simulability. This work will lead to more precise algorithms and is a step towards understanding how QC quantum advantage is affected by the underlying physics.
Boson Sampling is the problem of sampling from the same output probability distribution as a collection of indistinguishable single photons input into a linear interferometer. It has been shown that, subject to certain computational complexity conjectures, in general the problem is difficult to solve classically, motivating optical experiments aimed at demonstrating quantum computational ``supremacy.'' There are a number of challenges faced by such experiments, including the generation of indistinguishable single photons. We provide a quantum circuit that simulates Boson Sampling with arbitrarily distinguishable particles. This makes clear how distinguishabililty leads to decoherence in the standard quantum circuit model, allowing insight to be gained. At the heart of the circuit is the quantum Schur transform, which follows from a representation theoretic approach to the physics of distinguishable particles in first quantization. The techniques are quite general and have application beyond Boson Sampling.
Abstract Boson Sampling is a task that is conjectured to be computationally hard for a classical computer, but which can be efficiently solved by linear-optical interferometers with Fock state inputs. Significant advances have been reported in the last few years, with demonstrations of small- and medium-scale devices, as well as implementations of variants such as Gaussian Boson Sampling. Besides the relevance of this class of computational models in the quest for unambiguous experimental demonstrations of quantum advantage, recent results have also proposed the first applications for hybrid quantum computing. Here, we introduce the adoption of non-linear photon–photon interactions in the Boson Sampling framework, and analyze the enhancement in complexity via an explicit linear-optical simulation scheme. By extending the computational expressivity of Boson Sampling, the introduction of non-linearities promises to disclose novel functionalities for this class of quantum devices. Hence, our results are expected to lead to new applications of near-term, restricted photonic quantum computers.