Feedback stabilization of discrete-time quantum systems subject to non-demolition measurements with imperfections and delays

TL;DR

This paper develops a feedback control method for discrete-time quantum systems that ensures convergence to a target state despite measurement imperfections and delays, using a supermartingale-based Lyapunov function.

Contribution

It introduces a novel state feedback scheme that accounts for delays and measurement errors, ensuring almost sure convergence in quantum systems.

Findings

The control scheme guarantees convergence even with measurement imperfections.

Simulations and experimental data validate the effectiveness of the feedback method.

Application demonstrated on a cavity quantum electrodynamics system.

Abstract

We consider a controlled quantum system whose finite dimensional state is governed by a discrete-time nonlinear Markov process. In open-loop, the measurements are assumed to be quantum non-demolition (QND). The eigenstates of the measured observable are thus the open-loop stationary states: they are used to construct a closed-loop supermartingale playing the role of a strict control Lyapunov function. The parameters of this supermartingale are calculated by inverting a Metzler matrix that characterizes the impact of the control input on the Kraus operators defining the Markov process. The resulting state feedback scheme, taking into account a known constant delay, provides the almost sure convergence of the controlled system to the target state. This convergence is ensured even in the case where the filter equation results from imperfect measurements corrupted by random errors with conditional probabilities given as a left stochastic matrix. Closed-loop simulations corroborated by experimental data illustrate the interest of such nonlinear feedback scheme for the photon box, a cavity quantum electrodynamics system.