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arXiv:2412.18664 · Physical Review A

Laplace expansions and tree decompositions: A faster polytime algorithm for shallow nearest-neighbor boson sampling

Samo Novák, Raúl García−Patrón

Published Feb 6, 2026• 2 citations
arXivHTMLPDF

Abstract

In a boson sampling quantum optical experiment, we send n individual photons into an m -mode interferometer and measure the occupation pattern on the output. The statistics of this process depend on the permanent of a matrix representing the experiment, which is itself a #P-hard problem to compute, and this is the reason why ideal and fully general boson sampling is hard to simulate on a classical computer. We exploit the fact that, for a nearest-neighbor shallow circuit, i.e., depth D = O ( log m ) , one can adapt the algorithm by Clifford and Clifford [SODA '18 (2018), pp. 146–155] to exploit the sparsity of the shallow interferometer using an algorithm by Cifuentes and Parrilo [] that can efficiently compute a permanent of a structured matrix from a tree decomposition. Our algorithm generates a sample from a shallow circuit in time O ( n 2 2 ω ω 2 ) + O ( ω n 3 ) , where ω is the treewidth of the decomposition that satisfies ω ≤ 2 D for nearest-neighbor shallow circuits. The key difference in our work with respect to previous work using similar methods is the reuse of the structure of the tree decomposition, allowing us to adapt the Laplace expansion used by Clifford and Clifford, which removes a significant factor of m from the running time, especially as m > n 2 is a requirement of the original boson sampling proposal.

DOI
10.1103/l4zf-ls9v

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In a boson sampling quantum optical experiment, we send n individual photons into an m -mode interferometer and measure the occupation pattern on the output. The statistics of this process depend on the permanent of a matrix representing the experiment, which is itself a #P-hard problem to compute, and this is the reason why ideal and fully general boson sampling is hard to simulate on a classical computer. We exploit the fact that, for a nearest-neighbor shallow circuit, i.e., depth D = O ( log m ) , one can adapt the algorithm by Clifford and Clifford [SODA '18 (2018), pp. 146–155] to exploit the sparsity of the shallow interferometer using an algorithm by Cifuentes and Parrilo [] that can efficiently compute a permanent of a structured matrix from a tree decomposition. Our algorithm generates a sample from a shallow circuit in time O ( n 2 2 ω ω 2 ) + O ( ω n 3 ) , where ω is the treewidth of the decomposition that satisfies ω ≤ 2 D for nearest-neighbor shallow circuits. The key difference in our work with respect to previous work using similar methods is the reuse of the structure of the tree decomposition, allowing us to adapt the Laplace expansion used by Clifford and Clifford, which removes a significant factor of m from the running time, especially as m > n 2 is a requirement of the original boson sampling proposal.
We show that boson sampling with an arbitrary mmm-mode interferometer and n≤mn\le mn≤m single-photon inputs can be simulated to inverse-polynomial total-variation error by a logarithmic-depth qubit circuit with polynomially many qubits. The circuit uses Clifford+TTT gates, arbitrary qubit connectivity, and a single final measurement, and its family is logspace uniform. The key idea is to enlarge the optical system, decompose the resulting transformation into six quadratic shears, and distribute each mode over many submodes. This redistribution permits a fixed local occupation cutoff, after which local basis changes and parallel phase gates give the qubit circuit. Consequently, our result places boson sampling within shallow quantum computation.
Abstract Since its introduction boson sampling has been the subject of intense study in the world of quantum computing. In the context of Fock-state boson sampling, the task is to sample independently from the set of all n × n submatrices built from possibly repeated rows of a larger m × n complex matrix according to a probability distribution related to the permanents of the submatrices. Experimental systems exploiting quantum photonic effects can in principle perform the task at great speed. For classical computing, Aaronson and Arkhipov (2011) showed that exact boson sampling problem cannot be solved in polynomial time unless the polynomial hierarchy collapses to the third level. Indeed for a number of years the fastest known exact classical algorithm ran in O ( m + n − 1 n n 2 n ) time per sample, emphasising the potential speed advantage of quantum computation. The advantage was reduced by Clifford and Clifford (2018), who gave a significantly faster classical solution taking O ( n 2 n + poly ( m , n ) ) time and linear space, matching the complexity of computing the permanent of a single matrix when m is polynomial in n. We continue by presenting an algorithm for Fock boson sampling whose average-case time complexity is much faster when m is proportional to n. In particular, when m = n our algorithm runs in approximately O(n · 1.69 n ) time on average. This result further increases the problem size needed to establish quantum computational advantage via the Fock scheme of boson sampling.
Abstract Boson sampling (BS) is viewed to be an accessible quantum computing paradigm to demonstrate computational advantage compared to classical computers. In this context, the evolution of permanent calculation algorithms attracts a significant attention as the simulation of BS experiments involves the evaluation of vast number of permanents. For this reason, we generalize the Balasubramanian–Bax–Franklin–Glynn permanent formula, aiming to efficiently integrate it into the BS strategy of Clifford and Clifford (2020 Faster classical boson sampling). A reduction in simulation complexity originating from multiplicities in photon occupation was achieved through the incorporation of a n-ary Gray code ordering of the addends during the permanent evaluation. Implementing the devised algorithm on FPGA-based data-flow engines, we leverage the resulting tool to accelerate boson sampling simulations for up to 40 photons. Drawing samples from a 60-mode interferometer, the achieved rate averages around 80 s per sample, employing 4 FPGA chips. The developed design facilitates the simulation of both ideal and lossy boson sampling experiments.
Boson Sampling, a computational task believed to be classically hard to simulate, is expected to hold promise for demonstrating quantum computational advantage using near-term quantum devices. However, noise in experimental implementations poses a significant challenge, potentially rendering Boson Sampling classically simulable and compromising its classical intractability. Numerous studies have proposed classical algorithms that can efficiently simulate Boson Sampling under various noise models, particularly as noise rates increase with circuit depth. To address this challenge, we investigate the viability of achieving quantum computational advantage through Boson Sampling implemented with shallow-depth linear optical circuits. In particular, as the average-case hardness of estimating output probabilities of Boson Sampling is a crucial ingredient in demonstrating its classical intractability, we make progress on establishing the average-case hardness of Boson Sampling confined to logarithmic-depth regimes. We also obtain the average-case hardness for logarithmic-depth Fock-state Boson Sampling subject to lossy environments and for the logarithmic-depth Gaussian Boson Sampling. By providing complexity-theoretical backgrounds for the classical simulation hardness of logarithmic-depth Boson Sampling, we expect that our findings will mark a crucial step towards a more noise-tolerant demonstration of quantum advantage with shallow-depth Boson Sampling.
A general quantum circuit can be simulated classically in exponential time. If it has a planar layout, then a tensor-network contraction algorithm due to Markov and Shi has a runtime exponential in the square root of its size, or more generally exponential in the treewidth of the underlying graph. Separately, Gottesman and Knill showed that if all gates are restricted to be Clifford, then there is a polynomial time simulation. We combine these two ideas and show that treewidth and planarity can be exploited to improve Clifford circuit simulation. Our main result is a classical algorithm with runtime scaling asymptotically as nω/2 < n1.19 which samples from the output distribution obtained by measuring all n qubits of a planar graph state in given Pauli bases. Here ω is the matrix multiplication exponent. We also provide a classical algorithm with the same asymptotic runtime which samples from the output distribution of any constant-depth Clifford circuit in a planar geometry. Our work improves known classical algorithms with cubic runtime.A key ingredient is a mapping which, given a tree decomposition of some graph G , produces a Clifford circuit with a structure that mirrors the tree decomposition and which emulates measurement of the corresponding graph state. We provide a classical simulation of this circuit with the runtime stated above for planar graphs and otherwise ntω−1 where t is the width of the tree decomposition. Our algorithm incorporates two subroutines which may be of independent interest. The first is a matrix-multiplication-time version of the Gottesman-Knill simulation of multi-qubit measurement on stabilizer states. The second is a new classical algorithm for solving symmetric linear systems over F2 in a planar geometry, extending previous works which only applied to non-singular linear systems in the analogous setting.
Boson sampling is a sampling task proven to be hard to simulate efficiently using classical computers under plausible assumptions, which makes it an appealing candidate for quantum supremacy. However, due to a large noise rate for near-term quantum devices, it is still unclear whether those noisy devices maintain the quantum advantage for much larger quantum systems. Since the noise rate typically grows with the circuit depth, an alternative is to find evidence of simulation hardness at the shallow-depth quantum circuit. To find the evidence, one way is to identify the minimum depth required for the average-case hardness of approximating output probabilities, which is considered a necessary condition for the state-of-the-art technique to prove the simulation hardness of boson sampling. In this work, we analyze the output probability distribution of shallow-depth boson sampling for Fock-states and Gaussian states, and examine the limitation of the average-case hardness argument at this shallow-depth regime for geometrically local architectures. We propose a shallow-depth linear optical circuit architecture that can overcome the problems associated with geometrically local architectures. Our numerical results suggest that this architecture demonstrates possibilities of average-case hardness properties in a shallow-depth regime, through its resemblance to the global Haar-random boson sampling circuit. This result implies that the corresponding architecture has the potential to be utilized for scalable quantum supremacy with its shallow-depth boson sampling.