Equating quantum imaginary time evolution, Riemannian gradient flows, and stochastic implementations

Physical Review Research, vol. 8

Abstract

We identify quantum imaginary time evolution as a Riemannian gradient flow on the unitary group. We develop an upper bound for the error between the two evolutions that can be controlled through the step size of the Riemannian gradient descent that minimizes the energy of the system. We discuss implementations through adaptive quantum algorithms and present a stochastic Riemannian gradient descent algorithm in which each step is efficiently implementable on a quantum computer. We prove that for a sufficiently small step size, the stochastic evolution concentrates around the imaginary time evolution, thereby providing performance guarantees for cooling the system through stochastic Riemannian gradient descent. Finally, we show that Riemannian gradient descent can be understood as an approximation to quantum imaginary time evolution that does not employ classical optimization.

Authors 3

1. Nathan A. McMahon Aachen

   Leiden University

   Affiliation as printed

   Leiden University

2. Mahum Pervez

   Arizona State University

   Affiliation as printed

   Arizona State University

3. Christian Arenz

   Arizona State University

   Affiliation as printed

   Arizona State University