  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.
