Speeding up quantum Markov processes through lifting

TL;DR

This paper extends the concept of lifting to quantum Markov processes, providing bounds on convergence rates and demonstrating how lifting can accelerate mixing speeds, with applications to classical and quantum systems.

Contribution

It introduces a general lifting framework for quantum Markov dynamics, establishing convergence bounds and applying it to various classical and quantum processes.

Findings

Lifting can at most transition diffusive to ballistic mixing speeds.

Derived lower bounds for convergence rates of lifted quantum processes.

Constructed optimal lifts for several classical and quantum Markov processes.

Abstract

We generalize the concept of non-reversible lifts for reversible diffusion processes initiated by Eberle and Lorler (2024) to quantum Markov dynamics. The lifting operation, which naturally results in hypocoercive processes, can be formally interpreted as, though not restricted to, the reverse of the overdamped limit. We prove that the $L^2$ convergence rate of the lifted process is bounded above by the square root of the spectral gap of its overdamped dynamics, indicating that the lifting approach can at most achieve a transition from diffusive to ballistic mixing speeds. Further, using the variational hypocoercivity framework based on space-time Poincare inequalities, we derive a lower bound for the convergence rate of the lifted dynamics. These findings not only offer quantitative convergence guarantees for hypocoercive quantum Markov processes but also characterize the potential and limitations of accelerating the convergence through lifting. In addition, we develop an abstract lifting framework in the Hilbert space setting applicable to any symmetric contraction $C_0$-semigroup, thereby unifying the treatment of classical and quantum dynamics. As applications, we construct optimal lifts for various detailed balanced classical and quantum processes, including the symmetric random walk on a chain, the depolarizing semigroup, Schur multipliers, and quantum Markov semigroups on group von Neumann algebras.