Complexity Classification of Conjugated Clifford Circuits

TL;DR

This paper demonstrates that conjugated Clifford circuits, which are Clifford circuits conjugated by a single-qubit gate, can perform classically hard sampling tasks, extending the understanding of the computational power of restricted quantum circuits.

Contribution

It classifies the computational power of conjugated Clifford circuits, showing that non-Clifford conjugations lead to classically hard sampling problems.

Findings

Non-Clifford conjugating unitaries enable hard sampling tasks.

Hardness extends to constant additive error under plausible assumptions.

Closes gaps in understanding the computational power of restricted quantum gate sets.

Abstract

Clifford circuits -- i.e. circuits composed of only CNOT, Hadamard, and $\pi/4$ phase gates -- play a central role in the study of quantum computation. However, their computational power is limited: a well-known result of Gottesman and Knill states that Clifford circuits are efficiently classically simulable. We show that in contrast, "conjugated Clifford circuits" (CCCs) -- where one additionally conjugates every qubit by the same one-qubit gate $U$ -- can perform hard sampling tasks. In particular, we fully classify the computational power of CCCs by showing that essentially any non-Clifford conjugating unitary $U$ can give rise to sampling tasks which cannot be efficiently classically simulated to constant multiplicative error, unless the polynomial hierarchy collapses. Furthermore, by standard techniques, this hardness result can be extended to allow for the more realistic model of constant additive error, under a plausible complexity-theoretic conjecture. This work can be seen as progress towards classifying the computational power of all restricted quantum gate sets.