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Quantum Information and Measurement (QIM)

Simulating complex networks in phase space: Gaussian boson sampling

P. D. Drummond, Bogdan Opanchuk, Alexander S. Dellios, M. D. Reid

Published Jan 1, 2021• 2 citations
Publisher

Abstract

We show how phase-space simulations of Gaussian quantum states in a photonic network permit verification of measurable correlations of Gaussian boson sampling (GBS) quantum computers. Our results agree with experiments for up to 100-th order correlations, provided decoherence is included. We extend this to more than 16,000 modes, and describe how to simulate genuine multipartite entanglement.

DOI
10.1364/qim.2021.f1c.5

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We show how phase-space simulations of Gaussian quantum states in a photonic network permit verification of measurable correlations of Gaussian boson sampling (GBS) quantum computers. Our results agree with experiments for up to 100-th order correlations, provided decoherence is included. We extend this to more than 16,000 modes, and describe how to simulate genuine multipartite entanglement.
We show how phase-space simulations of quantum states in a linear photonic network permit the verification of measurable probabilities and entanglement. We compare our predictions with recent Gaussian boson sampling experiments of Zhong et al. These use squeezed inputs and efficient ``on-off'' detectors, with up to 76th-order measured coincidence counts in the data. We introduce a general definition of grouped ``on-off'' detection probabilities for this purpose. The positive-PPP phase-space method is used to compute any grouped or marginal click probabilities. Additional decoherence is included to obtain agreement between theory and experiment. The only limitation in estimating grouped probabilities is the computational sampling error, which is similar in magnitude to the experimental sampling error. The results obtained and graphed here are from first-order up to 16 000th-order grouped count probabilities. However, any order between these is also computable. We extend these results to include grouped probabilities with multidimensional outcomes that have a polynomial number of points. We also analyze quadrature detection experiments and show how to simulate genuine multipartite entanglement using Wigner phase-space methods.
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.
Computational validation is vital for large-scale quantum computers. One needs computers that are both fast and accurate along with tests that encapsulate the complexity of such networks. Here, we utilize positive-P phase-space simulations of grouped count probabilities (GCPs) as a fingerprint for verifying multi-mode data from large Gaussian boson sampling (GBS) quantum computers that claim quantum computational advantage. Such simulations allow one to distinguish between true non-classical correlations generated by an experiment, and those obtained from a classical algorithm that replicates the photo-detection measurements. We use these methods to verify the outputs of a 144-channel GBS experiment, where we show data is far enough from the ideal ground truth distribution that it is beaten by a classical algorithm implemented using purely classical states, although it fails against a best-fit ground truth that includes decoherence. This gives evidence consistent with quantum computational advantage for a modified problem: a non-classical, partly thermalized ground truth model. Even with this model, discrepancies are present, indicating possible parameter estimation errors. Randomly generated tests from multiple high-order grouped counts are also performed. Each of these can be measured and simulated, providing a verification method that may be hard to replicate classically.
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.