Possibilistic simulation of quantum circuits by classical circuits

Abstract

In a breakthrough work, Bravyi, Gosset, and K\"onig (BGK) [Science 362, 308 (2018)] unconditionally proved that constant-depth quantum circuits are more powerful than their classical counterparts. Their result is equivalent to saying that a particular family of constant-depth quantum circuits takes classical circuits at least $\mathrm{\ensuremath{\Omega}}(logn)$ depth to ``simulate,'' in a certain sense. In our paper, we formalize their sense of simulation, which we call ``possibilistic simulation'' or ``$p$-simulation,'' and construct explicit classical circuits that can $p$-simulate any depth-$d$ quantum circuit with Clifford and $t\phantom{\rule{4pt}{0ex}}T$-gates in depth $O(d+t)$. Our classical circuits use ${\mathrm{NOT},\mathrm{AND},\mathrm{OR}}$ gates of fan-in $\ensuremath{\le}2$.