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

TL;DR

This paper presents a simpler, deterministic quasi-polynomial time algorithm and a PTAS for approximating the permanent of random matrices with small mean, advancing the understanding of average-case complexity relevant to quantum supremacy.

Contribution

It improves previous algorithms by providing a deterministic quasi-polynomial time algorithm and a PTAS for matrices with mean at least 1/polylog(n), simplifying the approach and enhancing flexibility.

Findings

Achieved a deterministic quasi-polynomial time algorithm.

Developed a PTAS with polynomial time in matrix size.

Improved approximation for matrices with mean at least 1/polylog(n).

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 $1/\mathbf{\mathrm{polyloglog}} (n)$. 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 $1/\mathbf{\mathrm{polylog}}(n)$. We note that if it can be further improved to $1/\mathbf{\mathrm{poly}}(n)$, 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.