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

arXiv (Cornell University)

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 which 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.

Authors 3

1. Nathan A. McMahon Aachen

   Leiden University

   Affiliation as printed

   LIACS , Leiden University , Netherlands

2. Mahum Pervez Aachen applied Quantum algorithms (aQa)

   Leiden University · Arizona State University

   Affiliation as printed

   Applied Quantum Algorithms , Leiden University , Netherlands

   School of Electrical, Computer, and Energy Engineering , Arizona State University , Tempe , Arizona 85281 , USA

3. Christian Arenz

   Arizona State University

   Affiliation as printed

   School of Electrical, Computer, and Energy Engineering , Arizona State University , Tempe , Arizona 85281 , USA