Characterization of randomness in quantum circuits of continuous gate sets

TL;DR

This paper extends the characterization of randomness in quantum circuits with continuous symmetries, providing detailed derivations and a general framework for symmetric local random circuits and their asymptotic properties.

Contribution

It offers a comprehensive derivation of theorems for general symmetries and introduces a framework for analyzing symmetric local unitary gate sets in quantum circuits.

Findings

Explicit characterization of asymptotic unitary designs for various symmetries

Derivation of main theorems for general symmetry groups

Framework for analyzing finite sets of connected compact subgroups

Abstract

In the accompanying paper of arXiv:2408.13472, we have established the method of characterizing the maximal order of asymptotic unitary designs generated by symmetric local random circuits, and have explicitly specified the order in the cases of $\mathbb{Z}_2$, U(1), and SU(2) symmetries. Here, we provide full details on the derivation of the main theorems for general symmetry and for concrete symmetries. Furthermore, we consider a general framework where we have access to a finite set of connected compact unitary subgroups, which includes symmetric local unitary gate sets.