  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.
