Bridging Classical and Quantum Information Scrambling with the Operator Entanglement Spectrum

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.