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

Nathan Ramusat and Vincenzo Savona

Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

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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.

Quantum systems are always subject to the influence of their surrounding environment. This influence very often causes an irreversible evolution of the system over time, leading to a well defined steady state independently of the initial conditions. Simulating the properties of the steady state has the same computational complexity as the simulation of isolated quantum systems, which for large systems is typically intractable with classical computers.

To tackle this problem we develop a quantum algorithm, suitable to be executed on the next generation of quantum computers, that solves the equation for the steady state of an open quantum system. The algorithm brings an exponential speedup over classical computational approaches in most physically relevant cases, thus holding promise as the election tool for the study of complex open quantum systems on upcoming quantum hardware.

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Cited by

[1] Di Luo, Zhuo Chen, Juan Carrasquilla, and Bryan K. Clark, "Autoregressive Neural Network for Simulating Open Quantum Systems via a Probabilistic Formulation", arXiv:2009.05580.

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