The complexity of energy eigenstates as a mechanism for equilibration

TL;DR

This paper explores how the complexity of energy eigenstates influences the equilibration of isolated quantum systems, linking quantum circuit complexity to the system's ability to reach equilibrium.

Contribution

It establishes a statistical relationship between Hamiltonian eigenvector complexity and equilibration, assuming a conjecture about quantum circuit incompressibility.

Findings

Hamiltonians with super-quadratic complexity typically equilibrate

Sub-linear complexity Hamiltonians generally do not equilibrate

Provides a formula for equilibration time-scale based on level density Fourier transform

Abstract

Understanding the mechanisms responsible for the equilibration of isolated quantum many-body systems is a long-standing open problem. In this work we obtain a statistical relationship between the equilibration properties of Hamiltonians and the complexity of their eigenvectors, provided that a conjecture about the incompressibility of quantum circuits holds. We quantify the complexity by the size of the smallest quantum circuit mapping the local basis onto the energy eigenbasis. Specifically, we consider the set of all Hamiltonians having complexity C, and show that almost all such Hamiltonians equilibrate if C is super-quadratic with the system size, which includes the fully random Hamiltonian case in the limit C to infinity, and do not equilibrate if C is sub-linear. We also provide a simple formula for the equilibration time-scale in terms of the Fourier transform of the level density. Our results are statistical and, therefore, do not apply to specific Hamiltonians. Yet, they establish a fundamental link between equilibration and complexity theory.