Unitary designs from statistical mechanics in random quantum circuits

TL;DR

This paper analyzes how local random quantum circuits rapidly approximate unitary k-designs using a statistical mechanics approach, providing exact expressions and depth bounds for convergence.

Contribution

It introduces a statistical mechanical mapping to exactly compute the convergence to unitary designs and establishes near-optimal depth bounds for various k-values.

Findings

Random circuits form approximate 2-designs in O(n) depth.

Circuits form approximate k-designs in O(nk) depth, nearly optimal.

Analytical computation of second moment confirms known results.

Abstract

Random quantum circuits are proficient information scramblers and efficient generators of randomness, rapidly approximating moments of the unitary group. We study the convergence of local random quantum circuits to unitary $k$-designs. Employing a statistical mechanical mapping, we give an exact expression of the distance to forming an approximate design as a lattice partition function. In the statistical mechanics model, the approach to randomness has a simple interpretation in terms of domain walls extending through the circuit. We analytically compute the second moment, showing that random circuits acting on $n$ qudits form approximate 2-designs in $O(n)$ depth, as is known. Furthermore, we argue that random circuits form approximate unitary $k$-designs in $O(nk)$ depth and are thus essentially optimal in both $n$ and $k$. We can show this in the limit of large local dimension, but more generally rely on a conjecture about the dominance of certain domain wall configurations.