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.
Highlights
Open quantum systems are rapidly emerging as a major research field [1,2,3]
A similar goal is pursued in Ref. [35] where the time evolution of correlators is simulated, the perturbative scheme used for the dissipative dynamics results in a computational cost scaling exponentially with time, inappropriate to extrapolate the non-equilibrium steady state (NESS) in the long-time limit
We present here a simple numerical test of the quantum algorithm by simulating the quantum circuit for an elementary model of an open quantum system
Summary
Open quantum systems are rapidly emerging as a major research field [1,2,3]. On the fundamental side, they can display new classes of universal physical properties such as dissipative phase transitions and topological phases, while applications can span from novel paradigms of quantum simulators [4, 5] to the accurate modeling of noise in modern quantum computing platforms [6]. We propose a different approach consisting in the direct estimation of the averaged expectation values of observables on the NESS, without requiring the integration of the system dynamics. This is achieved by representing the density matrix in vectorized form and mapping the steady-state condition onto a linear system of equations. The initial state is prepared leveraging the known spectral properties of the generator of the open-system dynamics, so to obtain a large overlap with the target output state In this way, the success probability of a single QPE run is O(1). Appendix A develops the oracular part of the algorithm for the specific case of a dissipative transverse Ising spin model
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