Commuting Quantum Circuits with Few Outputs are Unlikely to be Classically Simulatable

TL;DR

This paper investigates the classical simulability of commuting quantum circuits with few outputs, providing evidence that such circuits are unlikely to be classically simulatable unless certain complexity-theoretic collapses occur.

Contribution

It demonstrates that commuting quantum circuits with exponentially few outputs are not classically simulatable unless the polynomial hierarchy collapses, and extends the analysis to Clifford circuits with non-Clifford layers.

Findings

Existence of non-simulatable commuting quantum circuits unless complexity collapses

Conditions under which generalized circuits are classically simulatable

Clifford circuits with a depth-1 non-Clifford layer are unlikely to be classically simulatable

Abstract

We study the classical simulatability of commuting quantum circuits with n input qubits and O(log n) output qubits, where a quantum circuit is classically simulatable if its output probability distribution can be sampled up to an exponentially small additive error in classical polynomial time. First, we show that there exists a commuting quantum circuit that is not classically simulatable unless the polynomial hierarchy collapses to the third level. This is the first formal evidence that a commuting quantum circuit is not classically simulatable even when the number of output qubits is exponentially small. Then, we consider a generalized version of the circuit and clarify the condition under which it is classically simulatable. Lastly, we apply the argument for the above evidence to Clifford circuits in a similar setting and provide evidence that such a circuit augmented by a depth-1 non-Clifford layer is not classically simulatable. These results reveal subtle differences between quantum and classical computation.