Riemannian geometry and automatic differentiation for optimization problems of quantum physics and quantum technologies

TL;DR

This paper introduces a Riemannian geometric approach combined with automatic differentiation to efficiently optimize complex quantum systems, enabling advancements in quantum state preparation, gate decomposition, and system control.

Contribution

It develops a unified Riemannian optimization framework for quantum problems, integrating automatic differentiation and providing open-source tools for broad quantum applications.

Findings

Effective optimization of multipartite Hamiltonians and tensor networks.

Successful quantum state preparation and gate decomposition.

Potential for controlling noisy quantum systems.

Abstract

Optimization with constraints is a typical problem in quantum physics and quantum information science that becomes especially challenging for high-dimensional systems and complex architectures like tensor networks. Here we use ideas of Riemannian geometry to perform optimization on manifolds of unitary and isometric matrices as well as the cone of positive-definite matrices. Combining this approach with the up-to-date computational methods of automatic differentiation, we demonstrate the efficacy of the Riemannian optimization in the study of the low-energy spectrum and eigenstates of multipartite Hamiltonians, variational search of a tensor network in the form of the multiscale entanglement-renormalization ansatz, preparation of arbitrary states (including highly entangled ones) in the circuit implementation of quantum computation, decomposition of quantum gates, and tomography of quantum states. Universality of the developed approach together with the provided open source software enable one to apply the Riemannian optimization to complex quantum architectures well beyond the listed problems, for instance, to the optimal control of noisy quantum systems.