In 2011, Aaronson gave a striking proof, based on quantum linear optics, showing that the problem of computing the permanent of a matrix is #P-hard. Aaronson's proof led naturally to hardness of approximation results for the permanent, and it was arguably simpler than Valiant's seminal proof of the same fact in 1979. Nevertheless, it did not prove that computing the permanent was #P-hard for any class of matrices which was not previously known. In this paper, we present a collection of new results about matrix permanents that are derived primarily via these linear optical techniques.
First, we show that the problem of computing the permanent of a real orthogonal matrix is #P-hard. Much like Aaronson's original proof, this will show that even a multiplicative approximation remains #P-hard to compute. The hardness result even translates to permanents over finite fields, where the problem of computing the permanent of an orthogonal matrix is ModpP-hard in the finite field F_{p^4} for all primes p not equal to 2 or 3. Interestingly, this characterization is tight: in fields of characteristic 2, the permanent coincides with the determinant; in fields of characteristic 3, one can efficiently compute the permanent of an orthogonal matrix by a nontrivial result of Kogan.
Finally, we use more elementary arguments to prove #P-hardness for the permanent of a positive semidefinite matrix, which shows that certain probabilities of boson sampling experiments with thermal states are hard to compute exactly despite the fact that they can be efficiently sampled by a classical computer.
Introduction to foundations of probability and randomness (for students in physics), Lectures given at the Institute of Quantum Optics and Quantum Information, Austrian Academy of Science, Lecture-1: Kolmogorov and von Mises
线性光学量子计算的一个基本问题是理解物理系统中量子霸权的起源。发现多模线性光学跃迁振幅是通过跃迁算子矩阵的积和来计算的，这是经典模拟中的一个难题（玻色子采样问题）。我们可以通过考虑直接决定计算跃迁振幅的运行时间的量子测度来理解这个问题。本文提出了一种量子度量--“Fock态并发和”$C_S$，它是对“广义Fock态并发”（一种类似于纠缠和相干的广义并发的度量）所有成员的求和.通过引入广义算法计算的Fock状态玻色子采样与任意数量的光子每模式的跃迁振幅，我们表明，所有已知的算法的最小经典运行时间直接依赖于$C_S$。因此，我们可以说，\emph{the Fock state concurrence sum $C_S$ behaves as a collective measure that controls the computational complexity of Fock state BS}。我们期望我们对Fock态并发在积算中的作用的观察将为解释线性光学的量子计算能力提供一个统一的观点。
