Non-Markovian Stochastic Schr\"odinger Equation: Matrix Product State Approach to the Hierarchy of Pure States

TL;DR

This paper introduces a numerically exact and efficient matrix product state approach for simulating non-Markovian open quantum systems at finite temperature, reducing computational complexity.

Contribution

It develops the HOMPS method by reformulating the stochastic hierarchy of pure states into a matrix product state framework, enabling scalable simulations.

Findings

Validated on the spin-boson model.

Efficient for long chains with non-Markovian environments.

Reduces complexity from exponential to polynomial.

Abstract

We derive a stochastic hierarchy of matrix product states (HOMPS) for non-Markovian dynamics in open quantum system at finite temperature, which is numerically exact and efficient. HOMPS is obtained from the recently developed stochastic hierarchy of pure states (HOPS) by expressing HOPS in terms of formal creation and annihilation operators. The resulting stochastic first order differential equation is then formulated in terms of matrix product states and matrix product operators. In this way the exponential complexity of HOPS can be reduced to scale polynomial with the number of particles. The validity and efficiency of HOMPS is demonstrated for the spin-boson model and long chains where each site is coupled to a structured, strongly non-Markovian environment.