Classical simulation of quantum computation, the Gottesman-Knill theorem, and slightly beyond

TL;DR

This paper presents a simplified proof of the Gottesman-Knill theorem, introduces a normal form for Clifford circuits, and explores the boundary between classically simulatable quantum circuits and those that are computationally hard to simulate.

Contribution

It provides a new, straightforward proof of the Gottesman-Knill theorem, introduces a normal form for Clifford circuits, and discusses the separation between weak and strong classical simulation.

Findings

Clifford circuits can be reduced to a normal form that is easily simulatable.

Weak simulation of certain quantum circuits is efficient, while strong simulation is #P-complete.

The normal form clarifies why Clifford circuits have limited computational power.

Abstract

We study classical simulation of quantum computation, taking the Gottesman-Knill theorem as a starting point. We show how each Clifford circuit can be reduced to an equivalent, manifestly simulatable circuit (normal form). This provides a simple proof of the Gottesman-Knill theorem without resorting to stabilizer techniques. The normal form highlights why Clifford circuits have such limited computational power in spite of their high entangling power. At the same time, the normal form shows how the classical simulation of Clifford circuits fits into the standard way of embedding classical computation into the quantum circuit model. This leads to simple extensions of Clifford circuits which are classically simulatable. These circuits can be efficiently simulated by classical sampling ('weak simulation') even though the problem of exactly computing the outcomes of measurements for these circuits ('strong simulation') is proved to be #P-complete--thus showing that there is a separation between weak and strong classical simulation of quantum computation.