Lanczos spectrum for random operator growth

TL;DR

This paper extends the application of Krylov methods to Heisenberg time evolution by tridiagonalizing the Liouvillian, providing analytical formulas verified through numerical experiments on Gaussian and non-Gaussian models.

Contribution

It introduces a method to tridiagonalize the Liouvillian in Heisenberg evolution, expanding the Krylov approach beyond Schrödinger evolution in quantum chaos studies.

Findings

Analytical formulas for Liouvillian tridiagonalization derived.

Numerical verification confirms formulas for Gaussian and non-Gaussian models.

Extension of Krylov methods to Heisenberg evolution demonstrated.

Abstract

Krylov methods have reappeared recently, connecting physically sensible notions of complexity with quantum chaos and quantum gravity. In these developments, the Hamiltonian and the Liouvillian are tridiagonalized so that Schrodinger/Heisenberg time evolution is expressed in the Krylov basis. In the context of Schrodinger evolution, this tridiagonalization has been carried out in Random Matrix Theory. We extend these developments to Heisenberg time evolution, describing how the Liouvillian can be tridiagonalized as well until the end of Krylov space. We numerically verify the analytical formulas both for Gaussian and non-Gaussian matrix models.