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 , and show that almost all such Hamiltonians equilibrate if is superquadratic in the system size, which includes the fully random Hamiltonian case in the limit , and do not equilibrate if is sublinear. 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.
- Received 19 October 2011
DOI:https://doi.org/10.1103/PhysRevE.87.032137
©2013 American Physical Society