Approximating Permanent of Random Matrices with Vanishing Mean: Made Better and Simpler

Abstract

The algorithm and complexity of approximating the permanent of a matrix is an extensively studied topic. Recently, its connection with quantum supremacy and more specifically BosonSampling draws special attention to the average-case approximation problem of the permanent of random matrices with zero or small mean value for each entry. Eldar and Mehraban (FOCS 2018) gave a quasi-polynomial time algorithm for random matrices with mean at least $1/\mathrm{polyloglog}(n)$. In this paper, we improve the result by designing a deterministic quasi-polynomial time algorithm and a PTAS for random matrices with mean at least $1/\mathrm{polylog}(n)$. We note that if it can be further improved to $1/\mathrm{poly}(n)$, it will disprove a central conjecture for quantum supremacy. Our algorithm is also much simpler and has a better and flexible trade-off for running time. The running time can be quasi-polynomial in both $n$ and $1/\epsilon$, or PTAS (polynomial in $n$ but exponential in $1/\epsilon$), where $\epsilon$ is the approximation parameter.

30 pages