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arXiv:2410.18740 · Physical Review A

Variational tensor network simulation of Gaussian boson sampling and beyond

Jonas Vinther, Michael J. Kastoryano

Published Jul 7, 2025• 0 citations
arXivHTMLPDF

Abstract

The continuous variable quantum computing platform constitutes a promising candidate for realizing quantum advantage, as exemplified in Gaussian Boson Sampling. While noise in the experiments makes the computation attainable for classical simulations, it has been suggested that the addition of non-linear elements to the experiment will help retain the quantum advantage. We propose a classical simulation tool for general continuous variable sampling problems, including Gaussian Boson Sampling and beyond. We reformulate the sampling problem as that of finding the ground state of a simple few-body Hamiltonian. This allows us to employ powerful variational methods based on tensor networks and to read off the simulation error directly from the expectation value of the Hamiltonian. We validate our method by simulating Gaussian Boson Sampling, where we achieve results comparable to the state of the art. We also consider a non-Gaussian sampling problem, for which we develop novel local basis optimization techniques based on a non-linear parameterization of the implicit basis, resulting in high effective cutoffs with diminished computational overhead.

DOI
10.1103/z463-7gqy

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The continuous variable quantum computing platform constitutes a promising candidate for realizing quantum advantage, as exemplified in Gaussian Boson Sampling. While noise in the experiments makes the computation attainable for classical simulations, it has been suggested that the addition of non-linear elements to the experiment will help retain the quantum advantage. We propose a classical simulation tool for general continuous variable sampling problems, including Gaussian Boson Sampling and beyond. We reformulate the sampling problem as that of finding the ground state of a simple few-body Hamiltonian. This allows us to employ powerful variational methods based on tensor networks and to read off the simulation error directly from the expectation value of the Hamiltonian. We validate our method by simulating Gaussian Boson Sampling, where we achieve results comparable to the state of the art. We also consider a non-Gaussian sampling problem, for which we develop novel local basis optimization techniques based on a non-linear parameterization of the implicit basis, resulting in high effective cutoffs with diminished computational overhead.
Tensor network is known as a powerful method to simulate quantum computation on a classical computer. Recently, the random quantum circuit used in the demonstration of quantum supremacy by Sycamore quantum computer have been simulated with tensor network method. Compared to the application of tensor networks to Discrete-Variable (DV) quantum computation, there are very few examples of application of tensor network to Continuous-Variable (CV) quantum computation. Therefore, we applied a tensor network structure called Matrix Product States (MPS) to CV quantum computation. Tensor network is expected to be useful for simulating Gaussian Boson Sampling (GBS), which is a promised near-term application of photonic quantum system. We outline the performance of tensor network-based simulation of CV quantum computation in GBS.
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.
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Abstract A growing cohort of experimental linear photonic networks implementing Gaussian boson sampling (GBS) have now claimed quantum advantage. However, many open questions remain on how to effectively verify these experimental results, as scalable methods are needed that fully capture the rich array of quantum correlations generated by these photonic quantum computers. In this paper, we briefly review recent theoretical methods to simulate experimental GBS networks. We focus mostly on methods that use phase-space representations of quantum mechanics, as these methods are highly scalable and can be used to validate experimental outputs and claims of quantum advantage for a variety of input states, ranging from the ideal pure squeezed vacuum state to more realistic thermalized squeezed states. A brief overview of the theory of GBS, recent experiments, and other types of methods are also presented. Although this is not an exhaustive review, we aim to provide a brief introduction to phase-space methods applied to linear photonic networks to encourage further theoretical investigations.
Quantum generative modeling has emerged as a promising application of quantum computers, aiming to model complex probability distributions beyond the reach of classical methods. In practice, however, training such models often requires costly gradient estimation performed directly on the quantum hardware. Crucially, for certain structured quantum circuits, expectation values of local observables can be efficiently evaluated on a classical computer, enabling classical training without calls to the quantum hardware in the optimization loop. In these models, sampling from the resulting circuits can still be classically hard, so inference must be performed on a quantum device, yielding a potential computational advantage. In this work, we introduce a photonic quantum generative model built on parametrized Gaussian Boson Sampling circuits. The training is based on the efficient classical evaluation of expectation values enabled by the Gaussian structure of the state, allowing scalable optimization of the model parameters through the maximum mean discrepancy loss function. We demonstrate the effectiveness of the approach through numerical experiments on photonic systems with up to 805 modes and over a million trainable parameters, highlighting its scalability and suitability for near-term photonic quantum devices.