A polynomial time additive estimate of the permanent using Gaussian fields

We present a polynomial-time randomized algorithm for estimating the permanent of an arbitrary $M \times M$ real matrix $A$ up to an additive error. We do this by viewing the permanent of $A$ as the expectation of a product of a centered joint Gaussian random variables whose covariance matrix we call the Gaussian embedding of $A$. The algorithm outputs the empirical mean $S_{N}$ of this product after sampling from this multivariate distribution $N$ times. In particular, after sampling $N$ samples, our algorithm runs in time $O(MN)$ with failure probability \begin{equation*} P(|S_{N}-\text{perm}(A)| > t) \leq \frac{3^{M}}{t^{2}N}\alpha^{2M} \end{equation*} for $\alpha \geq \|A \|$.