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

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 a 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/polyloglog(n). In this paper, we improve the result by designing a deterministic quasi-polynomial time algorithm and a PTAS for random matrices whose module of mean is at least 1/ polylog(n). We note that if the algorithm can be further improved to work with a mean value that is a sufficiently small 1/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/∊, or PTAS (polynomial in n but exponential in 1/∊), where ∊ is the approximation parameter.