Elements of naturality in dynamical simulation frameworks for Hamiltonian, thermostatic, and Lindbladian flows on classical and quantum state-spaces

TL;DR

This paper develops a comprehensive geometric framework for simulating various physical processes, including Hamiltonian, Lindbladian, and thermostatic flows, on classical and quantum state-spaces, with applications in quantum spin microscopy.

Contribution

It introduces a natural geometric framework with explicit criteria for simulating diverse dynamical processes on classical and quantum state-spaces, validated through examples and applications.

Findings

Framework encompasses Hamiltonian, Lindbladian, and thermostatic flows.

Validation criteria for metric and symplectic flows are provided.

Applications include quantum spin microscopy and dynamic nuclear polarization.

Abstract

The practical focus of this work is the dynamical simulation of polarization transport processes in quantum spin microscopy and spectroscopy. The simulation framework is built-up progressively, beginning with state-spaces (configuration manifolds) that are geometrically natural, introducing coordinates that are algebraically natural; and finally specifying dynamical potentials that are physically natural; in each respect explicit criteria are given for "naturality." The resulting framework encompasses Hamiltonian flow (both classical and quantum), quantum Lindbladian processes, and classical thermostatic processes. Constructive validation and verification criteria are given for metric and symplectic flows on classical, quantum, and hybrid state-spaces, with particular emphasis to tensor network state-spaces. Both classical and quantum examples are presented, including dynamic nuclear polarization (DNP). A broad span of applications and challenges is discussed, ranging from the design and simulation of quantum spin microscopes to the design and simulation of quantum oracles.