Abstract
In breakthrough work, Bravyi, Gosset, and K\"{o}nig (BGK) [Science, 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 Ω(log n) depth to "simulate", in a certain sense. In our paper, we formalise 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 T-gates in depth O(d+t). Our classical circuits use {NOT, AND, OR} gates of fan-in ≤ 2.