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arXiv · 1408.3712

What can quantum optics say about computational complexity theory?

Saleh Rahimi-Keshari·Austin P. Lund·Timothy C. Ralph
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Abstract

Considering the problem of sampling from the output photon-counting probability distribution of a linear-optical network for input Gaussian states, we obtain results that are of interest from both quantum theory and the computational complexity theory point of view. We derive a general formula for calculating the output probabilities, and by considering input thermal states, we show that the output probabilities are proportional to permanents of positive-semidefinite Hermitian matrices. It is believed that approximating permanents of complex matrices in general is a #P-hard problem. However, we show that these permanents can be approximated with an algorithm in BPP^NP complexity class, as there exists an efficient classical algorithm for sampling from the output probability distribution. We further consider input squeezed-vacuum states and discuss the complexity of sampling from the probability distribution at the output.

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Saleh Rahimi-Keshari, Austin P. Lund, Timothy C. Ralph. 2015-02-13. What can quantum optics say about computational complexity theory?. https://doi.org/10.1103/physrevlett.114.060501

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