Unitary Designs of Symmetric Local Random Circuits

TL;DR

This paper characterizes how symmetric local random circuits can generate unitary t-designs, linking symmetry, locality, and randomness, and provides explicit bounds for different symmetry groups.

Contribution

It introduces a method to determine when symmetric local random circuits form unitary t-designs and calculates maximal design orders for various symmetries.

Findings

Necessary and sufficient condition for asymptotic t-design formation

Explicit maximal order of unitary design for specific symmetries

Connection between symmetry, locality, and randomness in quantum circuits

Abstract

We have established the method of characterizing the unitary design generated by a symmetric local random circuit. Concretely, we have shown that the necessary and sufficient condition for the circuit asymptotically forming a t-design is given by simple integer optimization for general symmetry and locality. By using the result, we explicitly give the maximal order of unitary design under the $\mathbb{Z}_2$, U(1), and SU(2) symmetries for general locality. This work reveals the relation between the fundamental notions of symmetry and locality in terms of randomness.