Design of Strict Control-Lyapunov Functions for Quantum Systems with QND Measurements

TL;DR

This paper develops a systematic method using graph theory and Laplacian matrices to design feedback control laws that stabilize quantum systems with QND measurements, enabling deterministic state preparation.

Contribution

It introduces a novel approach for constructing strict control-Lyapunov functions for quantum systems with QND measurements, ensuring global stabilization and deterministic state preparation.

Findings

Successfully stabilizes quantum systems to target states

Provides a systematic design method using graph theory

Demonstrates effectiveness through simulations with photon counting

Abstract

We consider discrete-time quantum systems subject to Quantum Non-Demolition (QND) measurements and controlled by an adjustable unitary evolution between two successive QND measures. In open-loop, such QND measurements provide a non-deterministic preparation tool exploiting the back-action of the measurement on the quantum state. We propose here a systematic method based on elementary graph theory and inversion of Laplacian matrices to construct strict control-Lyapunov functions. This yields an appropriate feedback law that stabilizes globally the system towards a chosen target state among the open-loop stable ones, and that makes in closed-loop this preparation deterministic. We illustrate such feedback laws through simulations corresponding to an experimental setup with QND photon counting.