Variational Quantum Algorithm for Solving the Liouvillian Gap

TL;DR

This paper introduces a variational quantum algorithm that efficiently estimates the Liouvillian gap in open quantum systems, enabling better understanding of relaxation dynamics using near-term quantum hardware.

Contribution

It presents a novel variational approach utilizing the Choi-Jamiokowski isomorphism and a two-stage optimization scheme to accurately compute the Liouvillian gap.

Findings

Numerical simulations show high accuracy across various system sizes.

The method effectively handles degenerate steady states.

Robust convergence demonstrated in dissipative XXZ model.

Abstract

In open quantum systems, the Liouvillian gap characterizes the relaxation time toward the steady state. However, accurately computing this quantity is notoriously difficult due to the exponential growth of the Hilbert space and the non-Hermitian nature of the Liouvillian superoperator. In this work, we propose a variational quantum algorithm for efficiently estimating the Liouvillian gap. By utilizing the Choi-Jamiokowski isomorphism, we reformulate the problem as finding the first excitation energy of an effective non-Hermitian Hamiltonian. Our method employs variance minimization with an orthogonality constraint to locate the first excited state and adopts a two-stage optimization scheme to enhance convergence. Moreover, to address scenarios with degenerate steady states, we introduce an iterative energy-offset scanning technique. Numerical simulations on the dissipative XXZ model confirm the accuracy and robustness of our algorithm across a range of system sizes and dissipation strengths. These results demonstrate the promise of variational quantum algorithms for simulating open quantum many-body systems on near-term quantum hardware.