On the Noisy Road to Open Quantum Dynamics: The Place of Stochastic Hamiltonians

Stochastic evolution underpins several approaches to the dynamics of open quantum systems, such as random modulation of Hamiltonian parameters, the stochastic Schrödinger equation (SSE), and the stochastic Liouville equation (SLE). These approaches replace the explicit system-environment coupling with an effective system-only dynamics, where dissipative behavior emerges from ensemble averaging. Stochastic Hamiltonians, in particular, have long served as phenomenological tools in physical chemistry to include environmental effects without recourse to an explicit microscopic derivation. In this work, we aim at a self-contained and accessible presentation of these approaches to further elaborate on their common roots in essential concepts of stochastic calculus and to delineate the conditions under which they are equivalent. We also discuss how different formulations naturally lead to different numerical time-integration schemes, better suited for either classical simulation platforms, based on finite-difference approximations, or quantum algorithms, that employ random unitary maps. Our analysis aims at providing a unified perspective and actionable recipes for classical and quantum implementations of stochastic evolution in the simulation of open quantum systems.