Oliver Buerschaper

Random quantum processes play a central role both in the study of fundamental mixing processes in quantum mechanics related to equilibration, thermalisation and black hole scrambling, as well as in process design. In this work, we present a theory for continuous-time unitary evolutions originating from local Hamiltonians having time-fluctuating terms, reflecting a Brownian motion on the unitary group. By tying the mathematical description closely with the more established one of random quantum circuits, we present a unified picture for analyzing local random quantum processes. Much of the progress reported is of technical nature: in particular, by relying on 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. We also introduce tools for proving bounds on the rate of decoupling from an environment with random quantum processes.