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arXiv:2311.05355 · Scientific Reports

Speeding up the classical simulation of Gaussian boson sampling with limited connectivity

Tian-Yu Yang, Xiang‐Bin Wang

Published Apr 1, 2024• 1 citation
arXivHTMLPDF

Abstract

Abstract Gaussian boson sampling (GBS) plays a crucially important role in demonstrating quantum advantage. As a major imperfection, the limited connectivity of the linear optical network weakens the quantum advantage result in recent experiments. In this work, we introduce an enhanced classical algorithm for simulating GBS processes with limited connectivity. It computes the loop Hafnian of an $$n \times n$$ n×n symmetric matrix with bandwidthwin $$O(nw2^w)$$ O(nw2w) time. It is better than the previous fastest algorithm which runs in $$O(nw^2 2^w)$$ O(nw22w) time. This classical algorithm is helpful on clarifying how limited connectivity affects the computational complexity of GBS and tightening the boundary for achieving quantum advantage in the GBS problem.

DOI
10.1038/s41598-024-58136-1

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Abstract Gaussian boson sampling (GBS) plays a crucially important role in demonstrating quantum advantage. As a major imperfection, the limited connectivity of the linear optical network weakens the quantum advantage result in recent experiments. In this work, we introduce an enhanced classical algorithm for simulating GBS processes with limited connectivity. It computes the loop Hafnian of an n×nn \times nn×n n×n symmetric matrix with bandwidthwin O(nw2w)O(nw2^w)O(nw2w) O(nw2w) time. It is better than the previous fastest algorithm which runs in O(nw22w)O(nw^2 2^w)O(nw22w) O(nw22w) time. This classical algorithm is helpful on clarifying how limited connectivity affects the computational complexity of GBS and tightening the boundary for achieving quantum advantage in the GBS problem.
Identifying the boundary beyond which quantum machines provide a computational advantage over their classical counterparts is a crucial step in charting their usefulness. Gaussian boson sampling (GBS), in which photons are measured from a highly entangled Gaussian state, is a leading approach in pursuing quantum advantage. State-of-the-art GBS experiments that run in minutes would require 600 million years to simulate using the best preexisting classical algorithms. Here, we present faster classical GBS simulation methods, including speed and accuracy improvements to the calculation of loop hafnians. We test these on a ∼100,000-core supercomputer to emulate GBS experiments with up to 100 modes and up to 92 photons. This reduces the simulation time for state-of-the-art GBS experiments to several months, a nine-orders of magnitude improvement over previous estimates. Last, we introduce a distribution that is efficient to sample from classically and that passes a variety of GBS validation methods.
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.
We introduce an algorithm for the classical simulation of Gaussian boson sampling that is quadratically faster than previously known methods.The complexity of the algorithm is exponential in the number of photon pairs detected, not the number of photons, and is directly proportional to the time required to calculate a probability amplitude for a pure Gaussian state.The main innovation is to use auxiliary conditioning variables to reduce the problem of sampling to the computation of the pure-state probability amplitudes, for which the most computationally expensive step is the calculation of a loop hafnian.We implement and benchmark an improved loop-hafnian algorithm and show that it can be used to compute pure-state probabilities, the dominant step in the sampling algorithm, of events involving up to 50 photons in a single workstation, i.e., without the need of a supercomputer.
Gaussian Boson Sampling (GBS) is capable of solving certain classes of graph problems owing to the samples produced by such a device having a connection to the hafnian matrix function. In particular, a GBS device has been shown to provide an enhancement in the search of cliques -- or complete subgraphs -- in undirected weighted graphs over classical algorithms. A graph can be mapped to a GBS experiment by configuring the squeezing parameters of the input states and programming the linear optical network. In practice, limited squeezing and photon loss degrade the performance of the GBS device for max-clique search. In comparison, coherent states -- often considered a classical resource due to their Poissonian statistics -- can be readily prepared across many modes using an attenuated laser. In this paper, we report an enhancement of the success rate of GBS in finding maximum weighted cliques by adding displacements under lossy conditions or when a limited amount of squeezing is available. Moreover, we report that this enhancement can be scaled up to large graphs with limited resource overheads.
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.
Gaussian boson sampling (GBS) seeks quantum advantage by sampling photon-number patterns generated with squeezed inputs and passive linear optics. Many proposed GBS applications instead target function computation by applying functions to mode-resolved photon-number outcomes---a natural form of experimental postprocessing that produces classical outputs. Sampling hardness alone, however, does not determine the complexity of these tasks. By analyzing the irreducible decomposition of fixed-photon-number operator spaces, we prove that the expectation value of every such function in the average case over passive linear-optical networks can be classically evaluated for inputs with finite squeezing strength. We also provide a classical algorithm that estimates this value to inverse-polynomial additive error. The result provides new theoretical tools for analyzing linear-optical quantum systems, helps clarify the origin of current GBS hardness evidence, and inspires new applications of GBS with genuine quantum advantages.