# Possibilistic simulation of quantum circuits by classical circuits

## TL;DR

This paper formalizes the concept of possibilistic simulation of quantum circuits by classical circuits and constructs explicit classical circuits that can efficiently p-simulate certain quantum circuits with Clifford and T gates.

## Contribution

It introduces the formal notion of p-simulation and provides explicit classical circuits that simulate quantum circuits with Clifford and T gates in depth proportional to their quantum depth.

## Key findings

- Classical circuits can p-simulate quantum circuits with Clifford and T gates.
- P-simulation depth is proportional to quantum circuit depth plus T-gate count.
- Explicit classical circuits use simple gates like NOT, AND, OR with fan-in ≤ 2.

## 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 $\Omega(\log n)$ 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 $\{\text{NOT, AND, OR}\}$ gates of fan-in $\leq 2$.