Bridging Classical and Quantum Information Scrambling with the Operator Entanglement Spectrum

TL;DR

This paper investigates how the operator entanglement spectrum can distinguish between classical automaton and fully quantum chaotic dynamics, revealing a transition to universal quantum behavior when superposition gates are introduced.

Contribution

It demonstrates that the operator entanglement spectrum differentiates automaton from quantum dynamics and shows how adding superposition gates induces a transition to universal quantum chaos.

Findings

Operator entanglement spectrum under automaton dynamics follows Bernoulli matrix statistics.

Random unitary dynamics are governed by Gaussian matrix statistics.

A small number of superposition gates induces a transition to universal quantum chaos.

Abstract

Universal features of chaotic quantum dynamics underlie our understanding of thermalization in closed quantum systems and the complexity of quantum computations. Reversible automaton circuits, comprised of classical logic gates, have emerged as a tractable means to study such dynamics. Despite generating no entanglement in the computational basis, these circuits nevertheless capture many features expected from fully quantum evolutions. In this work, we demonstrate that the differences between automaton dynamics and fully quantum dynamics are revealed by the operator entanglement spectrum, much like the entanglement spectrum of a quantum state distinguishes between the dynamics of states under Clifford and Haar random circuits. While the operator entanglement spectrum under random unitary dynamics is governed by the eigenvalue statistics of random Gaussian matrices, we show evidence that under random automaton dynamics it is described by the statistics of Bernoulli random matrices, whose entries are random variables taking values $0$ or $1$. We study the crossover between automaton and generic unitary operator dynamics as the automaton circuit is doped with gates that introduce superpositions, namely Hadamard or $R_x = e^{-i\frac{\pi}{4}X}$ gates. We find that a constant number of superposition-generating gates is sufficient to drive the operator dynamics to the random-circuit universality class, similar to earlier results on Clifford circuits doped with $T$ gates. This establishes the operator entanglement spectrum as a useful tool for probing the chaoticity and universality class of quantum dynamics.