Effective field theory of random quantum circuits

TL;DR

This paper develops an effective field theory for random quantum circuits, revealing universal spectral statistics and providing tools to analyze quantum chaos and thermalization in many-body quantum systems.

Contribution

It introduces a replica sigma model framework for random quantum circuits, enabling derivation of universal random matrix behavior and rederivation of Weingarten calculus.

Findings

Reproduces Wigner-Dyson spectral statistics for various circuit models

Shows universal level statistics are preserved under permutations and higher dimensions

Provides a quantitative method to analyze quantum dynamics and chaos in Floquet systems

Abstract

Quantum circuits have been widely used as a platform to simulate generic quantum many-body systems. In particular, random quantum circuits provide a means to probe universal features of many-body quantum chaos and ergodicity. Some such features have already been experimentally demonstrated in the noisy intermediate-scale quantum (NISQ) devices. On the theory side, properties of random quantum circuits have been studied on a case-by-case basis and for certain specific systems, a hallmark of quantum chaos - universal Wigner-Dyson level statistics - has been derived. This work develops an effective field theory for a large class of random quantum circuits. The theory has the form of a replica sigma model and is similar to the low-energy approach to diffusion in disordered systems. The method is used to explicitly derive universal random matrix behavior of a large family of random circuits. In particular, we rederive Wigner-Dyson spectral statistics of the brickwork circuit model by Chan, De Luca, and Chalker [Phys. Rev. X 8, 041019 (2018)] and show within the same calculation that its various permutations and higher-dimensional generalizations preserve the universal level statistics. Finally, we use the replica sigma model framework to rederive the Weingarten calculus, which is a method to evaluate integrals of polynomials of matrix elements with respect to the Haar measure over compact groups and has many applications in the studies of quantum circuits. The effective field theory, derived here, provides both a method to quantitatively characterize quantum dynamics of random Floquet systems (e.g., calculating operator and entanglement spreading) and also path to understanding the general fundamental mechanism behind quantum chaos and thermalization in these systems.