Quantum gradient descent algorithms for nonequilibrium steady states and linear algebraic systems

TL;DR

This paper introduces quantum gradient descent algorithms tailored for simulating nonequilibrium steady states in open quantum systems and solving linear algebra problems, enhancing quantum computational methods.

Contribution

It develops efficient quantum algorithms for nonequilibrium steady states and linear algebra, utilizing Choi-Jamiolkowski isomorphism and adapting gradient descent for quantum simulations.

Findings

Algorithms successfully simulate steady states of dissipative quantum models.

Effective for solving linear systems and matrix-vector multiplications.

Numerical tests confirm the algorithms' efficiency and accuracy.

Abstract

The gradient descent approach is the key ingredient in variational quantum algorithms and machine learning tasks, which is an optimization algorithm for finding a local minimum of an objective function. The quantum versions of gradient descent have been investigated and implemented in calculating molecular ground states and optimizing polynomial functions. Based on the quantum gradient descent algorithm and Choi-Jamiolkowski isomorphism, we present approaches to simulate efficiently the nonequilibrium steady states of Markovian open quantum many-body systems. Two strategies are developed to evaluate the expectation values of physical observables on the nonequilibrium steady states. Moreover, we adapt the quantum gradient descent algorithm to solve linear algebra problems including linear systems of equations and matrix-vector multiplications, by converting these algebraic problems into the simulations of closed quantum systems with well-defined Hamiltonians. Detailed examples are given to test numerically the effectiveness of the proposed algorithms for the dissipative quantum transverse Ising models and matrix-vector multiplications.