Hierarchical Schr\"odinger Equations of Motion for Open Quantum Dynamics

TL;DR

This paper develops hierarchical Schrödinger equations of motion to accurately simulate open quantum system dynamics, especially non-Markovian effects, with reduced computational resources, demonstrated through quantum annealing of a ferromagnetic p-spin model.

Contribution

The paper introduces a novel hierarchical Schrödinger equations of motion framework based on the Feynman-Vernon influence functional for efficient simulation of non-Markovian quantum dynamics.

Findings

Effective simulation of non-Markovian quantum dynamics with reduced memory requirements.

Successful demonstration of the method on a quantum annealing process for a ferromagnetic p-spin model.

The approach can be tailored to various bath correlation functions and spectral densities.

Abstract

We rigorously investigate the quantum non-Markovian dissipative dynamics of a system coupled to a harmonic oscillator bath by deriving hierarchical Schrodinger equations of motion (HSEOM) and studying their dynamics. The HSEOM are the equations for wave functions derived on the basis of the Feynman-Vernon influence functional formalism for the density operator, $\langle q|\rho(t)|q' \rangle$, where $\langle q|$ and $|q' \rangle$ are the left- and right-hand elements. The time evolution of $\langle q|$ is computed from time $0$ to $t$, and subsequently, the time evolution of $|q' \rangle$ is computed from time $t$ to $0$ along a contour in the complex time plane. By appropriately choosing functions for the bath correlation function and the spectral density, we can take advantage of an HSEOM method to carry out simulations without the need for a great amount of computational memory. As a demonstration, quantum annealing simulation for a ferromagnetic $p$-spin model is studied.