Generalizing and derandomizing Gurvits's approximation algorithm for the permanent

Abstract

Around 2002, Leonid Gurvits gave a striking randomized algorithm to approximate the permanent of an $n\times n$ matrix A. The algorithm runs in $O( n^{2}/\varepsilon^{2})$ time, and approximates $\operatorname*{Per}( A)$ to within $\pm\varepsilon\left\Vert A\right\Vert ^{n}$ additive error. A major advantage of Gurvits's algorithm is that it works for arbitrary matrices, not just for nonnegative matrices. \ This makes it highly relevant to quantum optics, where the permanents of bounded-norm complex matrices play a central role. (In particular, $n\times n$ permanents arise as the transition amplitudes for $n$ identical photons.) Indeed, the existence of Gurvits's algorithm is why, in their recent work on the \textit{hardness} of quantum optics, Aaronson and Arkhipov (AA) had to talk about sampling problems rather than ordinary decision problems. In this paper, we improve Gurvits's algorithm in two ways. First, using an idea from quantum optics, we generalize the algorithm so that it yields a better approximation when the matrix A has either repeated rows or repeated columns. Translating back to quantum optics, this lets us classically estimate the probability of any outcome of an AA-type experiment -- even an outcome involving multiple photons "bunched" in the same mode -- at least as well as that probability can be estimated by the experiment itself. It also yields a general upper bound on the probabilities of "bunched" outcomes, which resolves a conjecture of Gurvits and might be of independent physical interest. Second, we use $\varepsilon$-biased sets to derandomize Gurvits's algorithm, in the special case where the matrix A is nonnegative. More interestingly, we generalize the notion of $\varepsilon$-biased sets to the complex numbers, construct "complex $\varepsilon$-biased sets", then use those sets to derandomize even our generalization of Gurvits's algorithm to the case (again for nonnegative A) where some rows or columns are identical. Whether Gurvits's algorithm can be derandomized for general A remains an outstanding problem.