Classical simulation complexity of extended Clifford circuits

TL;DR

This paper analyzes the classical simulation complexity of extended Clifford circuits, revealing many configurations are not efficiently simulatable and highlighting the close relationship between classical and quantum computational powers.

Contribution

It systematically studies the classical simulation complexity of various extended Clifford circuits, identifying which configurations are classically hard to simulate.

Findings

Many circuit configurations are not classically efficiently simulatable.

Certain classical extensions can turn classically simulatable circuits into universal quantum computers.

The results connect classical simulation complexity with quantum computational power.

Abstract

Clifford gates are a winsome class of quantum operations combining mathematical elegance with physical significance. The Gottesman-Knill theorem asserts that Clifford computations can be classically efficiently simulated but this is true only in a suitably restricted setting. Here we consider Clifford computations with a variety of additional ingredients: (a) strong vs. weak simulation, (b) inputs being computational basis states vs. general product states, (c) adaptive vs. non-adaptive choices of gates for circuits involving intermediate measurements, (d) single line outputs vs. multi-line outputs. We consider the classical simulation complexity of all combinations of these ingredients and show that many are not classically efficiently simulatable (subject to common complexity assumptions such as P not equal to NP). Our results reveal a surprising proximity of classical to quantum computing power viz. a class of classically simulatable quantum circuits which yields universal quantum computation if extended by a purely classical additional ingredient that does not extend the class of quantum processes occurring.