A GENERAL FIRST- AND SECOND-ORDER NUMERICAL SOLVER FOR NON-MARKOVIAN QUANTUM STATE DIFFUSION

Research Question and Approach

Numerical simulation of open quantum systems in non-Markovian regimes is notoriously difficult because the system's future state depends on its entire history through a memory kernel. The non-Markovian quantum state diffusion (NMQSD) equation captures this via a stochastic Schrödinger equation, but it involves functional derivatives with respect to the stochastic process that are computationally intractable. Existing methods, such as the hierarchy of pure states (HOPS), typically require the bath correlation function to be decomposed into a sum of exponentials. This paper addresses the need for a general numerical framework that does not rely on such decompositions.

The authors derive an analytical Dyson-type solution to the linear NMQSD equation, which reveals three fundamental structures: stochastic propagation, functional-derivative insertion, and memory pairing. By treating these structures as distinct operations, they construct a general auxiliary-state framework. This framework systematically separates the problem into time discretization, memory quadrature, and hierarchy truncation, allowing for the derivation of explicit first- and second-order numerical schemes.

The primary contribution is a robust, general-purpose numerical solver for NMQSD. The authors introduce a diagrammatic representation that maps the complex functional derivative terms into clear, step-by-step update rules. These rules allow for the explicit implementation of first- and second-order schemes that are applicable to arbitrary bath correlation functions and multi-level quantum systems. Numerical experiments verify that the proposed methods achieve the expected temporal accuracy, providing a flexible alternative to traditional hierarchical methods that are restricted by specific bath models.

This work significantly lowers the barrier for simulating non-Markovian quantum dynamics in complex environments. By removing the requirement for specific bath correlation decompositions, researchers can now apply stochastic wave function methods to a much broader class of physical systems, including those with structured or non-exponential memory kernels. The diagrammatic approach also provides a transparent way to visualize and implement these complex dynamics, making the numerical construction more accessible and easier to extend to higher-order accuracy.