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Classical algorithm for simulating experimental Gaussian boson sampling

Abstract

Gaussian boson sampling is a form of non-universal quantum computing that has been considered a promising candidate for showing experimental quantum advantage. While there is evidence that noiseless Gaussian boson sampling is hard to efficiently simulate using a classical computer, current Gaussian boson sampling experiments inevitably suffer from high photon loss rates and other noise sources. Nevertheless, they are currently claimed to be hard to classically simulate. Here we present a classical tensor-network algorithm that simulates Gaussian boson sampling and whose complexity can be significantly reduced when the photon loss rate is high. Our algorithm enables us to simulate the largest-scale Gaussian boson sampling experiment so far using relatively modest computational resources. We exhibit evidence that our classical sampler can simulate the ideal distribution better than the experiment can, which calls into question the claims of experimental quantum advantage.

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Fig. 1: Decomposition of lossy GBS circuit.
Fig. 2: Correlation of samples’ photon-number correlations to the ground-truth correlations.
Fig. 3: Required bond dimension for MPS truncation error.
Fig. 4: Computational cost estimation for larger systems.

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Data availability

Samples generated from our method can be found via Open Science Framework at https://doi.org/10.17605/OSF.IO/49TRH. Source data are provided with this paper.

Code availability

Code for our numerical simulation and data analysis can be found via Zenodo at https://doi.org/10.5281/zenodo.7992736.

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Acknowledgements

We thank Y.-H. Deng and C.-Y. Lu for providing the dataset of the Jiuzhang3.0 experiment and for interesting and fruitful discussions. We thank N. Quesada and J. Bulmer for helpful discussions. This research used the resources of the Argonne Leadership Computing Facility, which is a US Department of Energy (DOE) Office of Science User Facility supported under contract DE-AC02-06CH11357. We are also grateful for the support of the University of Chicago Research Computing Center for assistance with the numerical simulations reported in this paper. We acknowledge The Walrus Python library for the open source of GBS algorithms53 and the Strawberry Fields library54. M.L. acknowledges support from DOE Q-NEXT. Y.A. acknowledges support from the US Department of Energy Office of Science under contract DE-AC02-06CH11357 at Argonne National Laboratory and Defense Advanced Research Projects Agency (DARPA) under contract no. HR001120C0068. B.F. acknowledges support from AFOSR (FA9550-21-1-0008). This material is based upon work partially supported by the National Science Foundation under grant CCF-2044923 (CAREER) and by the US Department of Energy, Office of Science, National Quantum Information Science Research Centers (Q-NEXT), as well as by DOE QuantISED grant DE-SC0020360. L.J. acknowledges support from the ARO (W911NF-23-1-0077), ARO MURI (W911NF-21-1-0325), AFOSR MURI (FA9550-19-1-0399, FA9550-21-1-0209 and FA9550-23-1-0338), NSF (OMA-1936118, ERC-1941583, OMA-2137642, OSI-2326767 and CCF-2312755) and NTT Research, Packard Foundation (2020-71479).

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Contributions

C.O. and M.L. developed the theory, implemented the numerical experiments and wrote the paper. Y.A., B.F. and L.J. directed the research and developed the theory. All authors edited the paper.

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Correspondence to Changhun Oh.

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Extended data

Extended Data Fig. 1 Characteristics of the squeezed state Vp from the decomposition for single-mode cases.

(a) Relation between the actual squeezing parameter s and the input squeezing r for different transmission rate η. (b) Ratio of the actual squeezed photons to the total photons of the output state V. (c) and (d) Actual squeezing parameter and squeezed photon numbers when the input squeezing parameter is infinite. The dots represent the Borealis, Jiuzhang2.0, and Jiuzhang3.0’s circuit’s transmission rate and their largest actual squeezing and squeezed photons, assuming that infinite input squeezing is used.

Source data

Extended Data Fig. 2 Bipartite decomposition of GBS circuit for entanglement analysis.

(a) Any pure output state of a GBS circuit can be decomposed as (b) the product of two-mode squeezed vacuum (TMSV) states followed by local Gaussian unitary operations. (c) Thus, after tracing out the other system, each local system can be described by the product of thermal states followed by Gaussian unitary operation. The Gaussian unitary operations for each system are different in general.

Extended Data Fig. 3 Small-size experiment simulation results.

(a)(b) Example output probability distributions. (c) TVD and (d) XEB for different photon number sectors. Here, for the TVD we used the empirically obtained probability distribution with 1 million samples for each sector, and we used 10,000 samples for XEB for each sector. They clearly show the agreement between the XEB and TVD. The error bars are the standard deviation obtained by 1,000 bootstrapping resamples.

Source data

Extended Data Fig. 4 Simulation results of Borealis M = 72 case with the MPS algorithm.

(a) XEB; (b) two-point correlation with different bond dimensions χ = 120, 160, 200, 240. For the two-point correlation function calculation, we have used 1 million samples for all cases. The inset of (a) represents the total photon number distribution, and the shaded region is the sectors we used for XEB. The error bars are the standard deviation obtained by 1,000 bootstrapping resamples.

Source data

Supplementary information

Supplementary Information

Supplementary Sections 1–9 and Supplementary Figs. 1–4.

Supplementary Data

Source data for supplementary figures.

Source data

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Oh, C., Liu, M., Alexeev, Y. et al. Classical algorithm for simulating experimental Gaussian boson sampling. Nat. Phys. (2024). https://doi.org/10.1038/s41567-024-02535-8

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