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This paper is concerned with the Coherent Quantum Linear Quadratic Gaussian(CQLQG) control problem of finding a stabilizing measurement-free quantumcontroller for a quantum plant so as to minimize an infinite-horizon meansquare performance index for the fully quantum closed-loop system. Incomparison with the observation-actuation structure of classical controllers,the coherent quantum feedback is less invasive to the quantum dynamics andquantum information. Both the plant and the controller are open quantum systemswhose dynamic variables satisfy the canonical commutation relations (CCRs) of aquantum harmonic oscillator and are governed by linear quantum stochasticdifferential equations (QSDEs). In order to correspond to such oscillators,these QSDEs must satisfy physical realizability (PR) conditions, which areorganised as quadratic constraints on the controller matrices and reflect thepreservation of CCRs in time. The CQLQG problem is a constrained optimizationproblem for the steady-state quantum covariance matrix of the plant-controllersystem satisfying an algebraic Lyapunov equation. We propose a gradient descentalgorithm equipped with adaptive stepsize selection for the numerical solutionof the problem. The algorithm finds a local minimum of the LQG cost over theparameters of the Hamiltonian and coupling operators of a stabilizing PRquantum controller, thus taking the PR constraints into account. A convergenceanalysis of the proposed algorithm is presented. A numerical example of alocally optimal CQLQG controller design is provided to demonstrate thealgorithm performance.

A Gradient Descent Approach to Optimal Coherent Quantum LQG Controller Design