The Complexity of Bipartite Gaussian Boson Sampling

Daniel Grier1,2, Daniel J. Brod3, Juan Miguel Arrazola4, Marcos Benicio de Andrade Alonso3, and Nicolás Quesada5

1Institute for Quantum Computing, University of Waterloo, Canada
2Department of Computer Science and Engineering and Department of Mathematics, University of California, San Diego, US
3Instituto de Física, Universidade Federal Fluminense, Niterói, RJ, 24210-340, Brazil
4Xanadu, Toronto, ON, M5G 2C8, Canada
5Department of Engineering Physics, École Polytechnique de Montréal, Montréal, QC, H3T 1JK, Canada

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Gaussian boson sampling is a model of photonic quantum computing that has attracted attention as a platform for building quantum devices capable of performing tasks that are out of reach for classical devices. There is therefore significant interest, from the perspective of computational complexity theory, in solidifying the mathematical foundation for the hardness of simulating these devices. We show that, under the standard Anti-Concentration and Permanent-of-Gaussians conjectures, there is no efficient classical algorithm to sample from ideal Gaussian boson sampling distributions (even approximately) unless the polynomial hierarchy collapses. The hardness proof holds in the regime where the number of modes scales quadratically with the number of photons, a setting in which hardness was widely believed to hold but that nevertheless had no definitive proof.
Crucial to the proof is a new method for programming a Gaussian boson sampling device so that the output probabilities are proportional to the permanents of submatrices of an arbitrary matrix. This technique is a generalization of Scattershot BosonSampling that we call BipartiteGBS. We also make progress towards the goal of proving hardness in the regime where there are fewer than quadratically more modes than photons (i.e., the high-collision regime) by showing that the ability to approximate permanents of matrices with repeated rows/columns confers the ability to approximate permanents of matrices with no repetitions. The reduction suffices to prove that GBS is hard in the constant-collision regime.

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