Polynomial-time additive-error estimation of output probabilities for shallow quantum circuits

We give a deterministic classical algorithm that estimates $\mathrm{\mid }⟨x\mathrm{\mid }U\mathrm{\mid }{0}^n⟩{\mathrm{\mid }}^2$ to additive error $\epsilon$ in $\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}(n,1\mathrm{/}\epsilon )$ time, where $U$ is a constant-depth quantum circuit comprised of gates with bounded fan-in and arbitrary connectivity, and $x$ is an arbitrary $n$-bit output string. This improves over prior state-of-the-art algorithms that takes $n^{O(log(n))}$ time for the same task, $n^{O(log(log(n))}$ when $U$ is geometrically local, and $n^{O(1)}$ for 2D geometrically-local circuits.