  This paper is concerned with coherent quantum linear quadratic Gaussian
(CQLQG) control. The problem is to find a stabilizing measurement-free quantum
controller for a quantum plant so as to minimize a mean square cost for the
fully quantum closed-loop system. The plant and controller are open quantum
systems interconnected through bosonic quantum fields. In comparison with the
observation-actuation structure of classical controllers, coherent quantum
feedback is less invasive to the quantum dynamics. The plant and controller
variables satisfy the canonical commutation relations (CCRs) of a quantum
harmonic oscillator and are governed by linear quantum stochastic differential
equations (QSDEs). In order to correspond to such oscillators, these QSDEs must
satisfy physical realizability (PR) conditions in the form of quadratic
constraints on the state-space matrices, reflecting the CCR preservation in
time. The symmetry of the problem is taken into account by introducing
equivalence classes of coherent quantum controllers generated by symplectic
similarity transformations. We discuss a modified gradient flow, which is
concerned with norm-balanced realizations of controllers. A line-search
gradient descent algorithm with adaptive stepsize selection is proposed for the
numerical solution of the CQLQG control problem. The algorithm finds a local
minimum of the LQG cost over the parameters of the Hamiltonian and coupling
operators of a stabilizing coherent quantum controller, thus taking the PR
constraints into account. A convergence analysis of the algorithm is presented.
Numerical examples of designing locally optimal CQLQG controllers are provided
in order to demonstrate the algorithm performance.

Quantum Physics;Systems and Control;Optimization and Control