A Numerical Approach to Optimal Coherent Quantum LQG Controller Design Using Gradient Descent

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.