Mixing properties of stochastic quantum Hamiltonians

TL;DR

This paper develops a framework for understanding how stochastic quantum Hamiltonians lead to rapid mixing and approximate unitary designs, with implications for quantum information processing and thermalization.

Contribution

It introduces bounds on mixing times for stochastic Hamiltonian evolutions and derives a k-independent local moment operator, advancing the analysis of quantum designs and expanders.

Findings

Bounds on mixing times for stochastic Hamiltonian evolutions

Analytical expression for local k-th moment operator independent of k

Unified analysis of local random quantum circuits and dissipative processes

Abstract

Random quantum processes play a central role both in the study of fundamental mixing processes in quantum mechanics related to equilibration, thermalisation and fast scrambling by black holes, as well as in quantum process design and quantum information theory. In this work, we present a framework describing the mixing properties of continuous-time unitary evolutions originating from local Hamiltonians having time-fluctuating terms, reflecting a Brownian motion on the unitary group. The induced stochastic time evolution is shown to converge to a unitary design. As a first main result, we present bounds to the mixing time. By developing tools in representation theory, we analytically derive an expression for a local k-th moment operator that is entirely independent of k, giving rise to approximate unitary k-designs and quantum tensor product expanders. As a second main result, we introduce tools for proving bounds on the rate of decoupling from an environment with random quantum processes. By tying the mathematical description closely with the more established one of random quantum circuits, we present a unified picture for analysing local random quantum and classes of Markovian dissipative processes, for which we also discuss applications.