### Abstract

We study the classical complexity of the exact Boson Sampling problem where the objective is to produce provably correct random samples from a particular quantum mechanical distribution. The computational framework was proposed in STOC ’11 by Aaronson and Arkhipov in 2011 as an attainable demonstration of ‘quantum supremacy’, that is a practical quantum computing experiment able

to produce output at a speed beyond the reach of classical (that is non-quantum) computer hardware. Since its introduction Boson Sampling has been the subject of intense international research in the world of quantum computing. On the face of it, the problem is challenging for classical computation.

Aaronson and Arkhipov show that exact Boson Sampling is not efficiently solvable by a classical computer unless P#P = BPPNP and the polynomial hierarchy collapses to the third level.

The fastest known exact classical algorithm for the standard Boson Sampling problem requires O((m+n−1n) n2n) time to produce samples for a system with input size n and m output modes, making it infeasible for anything but the smallest values of n and m. We give an algorithm that is much faster, running in O(n2n + poly(m, n)) time and O(m) additional space. The algorithm is simple to

implement and has low constant factor overheads. As a consequence our classical algorithm is able to solve the exact Boson Sampling problem for system sizes far beyond current photonic quantum computing experimentation, thereby significantly reducing the likelihood of achieving near-term quantum supremacy in the context of Boson Sampling.

to produce output at a speed beyond the reach of classical (that is non-quantum) computer hardware. Since its introduction Boson Sampling has been the subject of intense international research in the world of quantum computing. On the face of it, the problem is challenging for classical computation.

Aaronson and Arkhipov show that exact Boson Sampling is not efficiently solvable by a classical computer unless P#P = BPPNP and the polynomial hierarchy collapses to the third level.

The fastest known exact classical algorithm for the standard Boson Sampling problem requires O((m+n−1n) n2n) time to produce samples for a system with input size n and m output modes, making it infeasible for anything but the smallest values of n and m. We give an algorithm that is much faster, running in O(n2n + poly(m, n)) time and O(m) additional space. The algorithm is simple to

implement and has low constant factor overheads. As a consequence our classical algorithm is able to solve the exact Boson Sampling problem for system sizes far beyond current photonic quantum computing experimentation, thereby significantly reducing the likelihood of achieving near-term quantum supremacy in the context of Boson Sampling.

Original language | English |
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Title of host publication | Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA18) |

Editors | Artur Czumaj |

Publisher | Society for Industrial and Applied Mathematics |

Pages | 146-155 |

Number of pages | 10 |

ISBN (Electronic) | 9781611975031 |

DOIs | |

Publication status | Published - 7 Jan 2018 |

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## Cite this

Clifford, P., & Clifford, R. (2018). The classical complexity of Boson sampling. In A. Czumaj (Ed.),

*Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA18)*(pp. 146-155). Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611975031.10