Model-Free Quantum Stabilization via Finite-Difference Lyapunov Control

TL;DR

This paper introduces a model-free quantum stabilization method that uses empirical finite-difference Lyapunov evaluations, requiring no prior knowledge of system dynamics, and guarantees asymptotic or practical stability.

Contribution

It presents a novel, derivative-free control framework for quantum state stabilization based solely on measurement data, applicable to arbitrary finite-dimensional systems.

Findings

Guarantees asymptotic stabilization in drift-free case

Achieves practical input-to-state stability with unknown drift and noise

Demonstrates effectiveness on a single qubit example

Abstract

We develop a model-free framework for stabilizing quantum states using only empirical finite-difference evaluations of a measurement-derived Lyapunov observable. The controller requires no knowledge of the Hamiltonian, dissipative structure, or generator of the dynamics, and relies solely on discrete measurement data. The approach combines three key elements: sign-based Lyapunov descent, adaptive gain amplification, and a finite-difference analogue of LaSalle's invariance principle. We provide rigorous conditions under which these mechanisms guarantee asymptotic stabilization along the sampling instants in the drift-free case and practical input-to-state stability (ISS) in the presence of unknown drift and noise. The resulting feedback law is simple, derivative-free, and experimentally feasible. A qubit example illustrates the complete closed-loop scheme and the predicted ISS-type behavior. Although demonstrated on a single qubit, the theory applies to arbitrary finite-dimensional quantum systems and offers a foundation for further developments in stochastic, subspace, and multi-qudit model-free quantum control.