Coherent quantum LQG controllers with Luenberger dynamics

TL;DR

This paper develops a framework for designing coherent quantum LQG controllers with Luenberger dynamics, addressing physical realizability and optimality conditions for quantum harmonic oscillators in a measurement-free setting.

Contribution

It introduces a novel approach combining quantum physical constraints with classical control structures to derive optimality conditions for quantum controllers.

Findings

Derived quadratic constraints on controller gain matrices.

Formulated algebraic equations for optimal controller design.

Linked quantum physical realizability with classical control principles.

Abstract

This paper is concerned with the coherent quantum linear-quadratic-Gaussian control problem of minimising an infinite-horizon mean square cost for a measurement-free field-mediated interconnection of a quantum plant with a stabilising quantum controller. The plant and the controller are multimode open quantum harmonic oscillators, governed by linear quantum stochastic differential equations and coupled to each other and the external multichannel bosonic fields in the vacuum state. We discuss an interplay between the quantum physical realizability conditions and the Luenberger structure associated with the classical separation principle. This leads to a quadratic constraint on the controller gain matrices, which is formulated in the framework of a swapping transformation for the conjugate positions and momenta in the canonical representation of the controller variables. For the class of coherent quantum controllers with the Luenberger dynamics, we obtain first-order necessary conditions of optimality in the form of algebraic equations, involving a matrix-valued Lagrange multiplier.