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arXiv:2502.19336

Using Gaussian Boson Samplers to Approximate Gaussian Expectation Problems

Jørgen Ellegaard Andersen, Shan Shan

Published Feb 26, 2025• 0 citations
arXivHTMLPDF

Abstract

Gaussian Boson Sampling (GBS) have shown advantages over classical methods for performing some specific sampling tasks. To fully harness the computational power of GBS, there has been great interest in identifying their practical applications. In this study, we explore the use of GBS samples for computing a numerical approximation to the Gaussian expectation problem, that is to integrate a multivariate function against a Gaussian distribution. We propose two estimators using GBS samples, and show that they both can bring an exponential speedup over the plain Monte Carlo (MC) estimator. Precisely speaking, the exponential speedup is defined in terms of the guaranteed sample size for these estimators to reach the same level of accuracy $ε$ and the same success probability $δ$ in the $(ε, δ)$ multiplicative error approximation scheme. We prove that there is an open and nonempty subset of the Gaussian expectation problem space for such computational advantage.

DOI
10.48550/arxiv.2502.19336

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Gaussian Boson Sampling (GBS), which can be realized with a photonic quantum computing model, perform some special kind of sampling tasks. In [4], we introduced algorithms that use GBS samples to approximate Gaussian expectation problems. We found a non-empty open subset of the problem space where these algorithms achieve exponential speedup over the standard Monte Carlo (MC) method. This speedup is defined in terms of the guaranteed sample size to reach the same accuracy εεε and success probability δδδ under the (ε,δ)(ε, δ)(ε,δ) multiplicative error approximation scheme. In this paper, we enhance our original approach by optimizing the average photon number in the GBS distribution to match the specific Gaussian expectation problem. We provide updated estimates of the guaranteed sample size for these improved algorithms and quantify the proportion of problem space where they outperform MC. Numerical results indicate that the proportion of the problem space where our improved algorithms have an advantage is substantial, and the advantage gained is significant. Notably, for certain special cases, our methods consistently outperform MC across nearly 100\% of the problem space.
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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.
Abstract Gaussian Boson Sampling is gaining relevance among the multiple technologies currently implementing quantum computing. The problem of sampling out of a boson distribution across multiple modes has a strong mathematical connection with combinatorics problems, with multiple practical applications that would classically be unfeasible to solve within reasonable time. The software stack that accompanies this GBS approach, and that takes the application from its high level description to the hardware implementation, is the main focus of this paper. The general compilation process of Gaussian Boson Samplers is described in detail. The specifics of the Strawberry Fields (Xanadu) compilation and simulation framework are discussed, along with its time profiling, and time implications on computationally significant problem sizes. Last, a compilation step to reduce the hardware description complexity is presented, demonstrating a linear reduction on the overall number of operators when the number of Gaussian operators increases.
Gaussian Boson Sampling (GBS) generates photon-count patterns whose probabilities are governed by hafnian functionals of an encoded covariance. Recent theory shows that GBS samples can approximate Gaussian expectation problems with provable sample-complexity advantages over Monte Carlo (MC) on an open, sizable subset of instances. We give a unifying explanation via the Deterministic Statistical Feedback Law (DSFL). First, GBS acts as a calibrated, admissible map that intertwines target statistics and physical sampling. Second, tuning the mean photon number is a genuine calibration step that minimizes a DSFL residual. Third, principal-angle (Friedrichs) geometry quantifies conditioning and leakage between statistical and sampling features. Fourth, a Lyapunov envelope provides non-asymptotic error decay and design rules. From these ingredients we derive bounds that recover and sharpen known GBS scalings, predict regimes where GBS outperforms MC, and specify a practical hybrid policy that switches by degree. Experiments with simulated distributions and hardware-aligned kernels support the theory and validate the tuning and selection rules.