Local random quantum circuits: ensemble CP maps and swap algebras

TL;DR

This paper introduces a formalism using ensemble CP maps and swap algebras to analyze local random quantum circuits, providing exact solutions and bounds on purity dynamics in various cases.

Contribution

It develops a novel formalism linking statistical properties of local random quantum circuits to completely positive maps and swap algebras, with exact solutions for 1D cases.

Findings

Exact solvability for 1D case illustrates the formalism.

Proves short-time area-law bounds on average purity.

Provides infinite-time results for uncorrelated and correlated circuits.

Abstract

We define different classes of local random quantum circuits (L-RQC) and show that: a) statistical properties of L-RQC are encoded into an associated family of completely positive maps and b) average purity dynamics can be described by the action of these maps on operator algebras of permutations (swap algebras). An exactly solvable one-dimensional case is analyzed to illustrate the power of the swap algebra formalism. More in general, we prove short time area-law bounds on average purity for uncorrelated L-RQC and infinite time results for both the uncorrelated and correlated cases.