Quantum algorithm for the simulation of open-system dynamics and thermalization

Abstract

The quantum open-system simulation is an important category of quantum simulation. By simulating the thermalization process at zero temperature, we can solve the ground-state problem of quantum systems. To realize the open-system evolution on the quantum computer, we need to encode the environment using qubits. However, usually the environment is much larger than the system, i.e., numerous qubits are required if the environment is directly encoded. In this paper, we propose a way to simulate open-system dynamics by reproducing reservoir correlation functions using a minimized Hilbert space. In this way, we only need a small number of qubits to represent the environment. To simulate the $n\mathrm{th}\ensuremath{-}\mathrm{order}$ expansion of the time-convolutionless master equation by reproducing up to $n$-time correlation functions, the number of qubits representing the environment is $\ensuremath{\sim}\ensuremath{\lfloor}\frac{n}{2}\ensuremath{\rfloor}{log}_{2}({N}_{\ensuremath{\omega}}{N}_{\ensuremath{\beta}})$. Here, ${N}_{\ensuremath{\omega}}$ is the number of frequencies in the discretized environment spectrum, and ${N}_{\ensuremath{\beta}}$ is the number of terms in the system-environment interaction. By reproducing two-time correlation functions, i.e., taking $n=2$, we can simulate the Markovian quantum master equation. In our algorithm, the environment on the quantum computer could be even smaller than the system.