A graph-theoretic approach to chaos and complexity in quantum systems

TL;DR

This paper introduces a graph-theoretic framework using commutator graphs to analyze chaos and complexity in quantum systems, providing finer insights than traditional dynamical Lie algebra methods.

Contribution

It develops a novel approach linking commutator graph properties to quantum chaos measures, enhancing the understanding of short-time dynamics in quantum systems.

Findings

Reduced average OTOC calculation to a graph counting problem

Connected Krylov complexity to the module structure of the DLA

Established a lower bound on average Krylov complexity based on shortest path length

Abstract

There has recently been considerable interest in studying quantum systems via dynamical Lie algebras (DLAs) -- Lie algebras generated by the terms which appear in the Hamiltonian of the system. However, there are some important properties that are revealed only at a finer level of granularity than the DLA. In this work we explore, via the commutator graph, average notions of scrambling, chaos and complexity over ensembles of systems with DLAs that possess a basis consisting of Pauli strings. Unlike DLAs, commutator graphs are sensitive to short-time dynamics, and therefore constitute a finer probe to various characteristics of the corresponding ensemble. We link graph-theoretic properties of the commutator graph to the out-of-time-order correlator (OTOC), the frame potential, the frustration graph of the Hamiltonian of the system, and the Krylov complexity of operators evolving under the dynamics. For example, we reduce the calculation of average OTOCs to a counting problem on the graph; separately, we connect the Krylov complexity of an operator to the module structure of the adjoint action of the DLA on the space of operators in which it resides, and prove that its average over the ensemble is lower bounded by the average shortest path length between the initial operator and the other operators in the commutator graph.