Operator growth in random quantum circuits with symmetry

TL;DR

This paper investigates how symmetry constraints in random quantum circuits affect operator growth, deriving transition probabilities, butterfly velocities, and diffusion constants to understand chaotic dynamics.

Contribution

It introduces a framework for analyzing operator growth in symmetric random circuits, providing explicit calculations of key dynamical quantities.

Findings

Derived transition probabilities for operator growth in symmetric circuits

Computed butterfly velocities for different symmetry classes

Calculated diffusion constants indicating operator spreading rates

Abstract

We study random quantum circuits with symmetry, where the local 2-site unitaries are drawn from a quotient or subgroup of the full unitary group $U(d)$. Random quantum circuits are minimal models of local quantum chaotic dynamics and can be used to study operator growth and the emergence of diffusive hydrodynamics. We derive the transition probabilities for the stochastic process governing the growth of operators in four classes of symmetric random circuits. We then compute the butterfly velocities and diffusion constants for a spreading operator by solving a simple random walk in each class of circuits.