A quantum algorithm for the direct estimation of the steady state of open quantum systems

TL;DR

This paper introduces a quantum algorithm that efficiently estimates the steady state of open quantum systems directly, avoiding long-time simulations and offering exponential advantages over classical methods.

Contribution

The authors develop a quantum algorithm that directly computes steady state expectation values without simulating full dynamics, utilizing quantum phase estimation on the Liouvillian's eigenvector.

Findings

Exponential speedup over classical diagonalization methods.

Efficient estimation of steady state observables.

Applicable away from critical points with small Liouvillian gap.

Abstract

Simulating the dynamics and the non-equilibrium steady state of an open quantum system are hard computational tasks on conventional computers. For the simulation of the time evolution, several efficient quantum algorithms have recently been developed. However, computing the non-equilibrium steady state as the long-time limit of the system dynamics is often not a viable solution, because of exceedingly long transient features or strong quantum correlations in the dynamics. Here, we develop an efficient quantum algorithm for the direct estimation of averaged expectation values of observables on the non-equilibrium steady state, thus bypassing the time integration of the master equation. The algorithm encodes the vectorized representation of the density matrix on a quantum register, and makes use of quantum phase estimation to approximate the eigenvector associated to the zero eigenvalue of the generator of the system dynamics. We show that the output state of the algorithm allows to estimate expectation values of observables on the steady state. Away from critical points, where the Liouvillian gap scales as a power law of the system size, the quantum algorithm performs with exponential advantage compared to exact diagonalization.