A numerical approach to optimal coherent quantum LQG controller design using gradient descent

Abstract

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.

Optimal Quantum Control Theory

“…A basic issue is the lack of a coherent filtering theory that is compatible with quantum mechanics. However, some research on parameterized approaches have appeared in the literature (e.g., 32,33).…”

A Karhunen-Loeve Expansion for One-mode Open Quantum Harmonic Oscillators Using the Eigenbasis of the Two-point Commutator Kernel

“…Also, θ > 0 is a risk-sensitivity parameter which is assumed to be sufficiently small to ensure that Ξ < +∞. In view of the asymptotic behaviour lim θ→0+ ln Ξ θ = EQ, the QEF Ξ extends the mean square cost EQ, which is used, for example, in coherent quantum LQG control problems [16], [19], [29], [31]. For finite values of θ > 0, the cost functional Ξ imposes an exponential penalty on the past history of the system variables captured by Q in a quadratic fashion.…”

Directly coupled observers for quantum harmonic oscillators with discounted mean square cost functionals and penalized back-action

“…for some A ∈ R p×p , where the matrix S is given by (99). In combination with (46), the relation (136) leads to the ODĖ…”

“…Fully quantum variational techniques, using perturbation analysis [45,54,55,57] beyond the class of OQHOs and symplectic geometric tools [46], suggest that the complicated sets of nonlinear equations for optimal quantum controllers and filters may appear to be more amenable to solution if they are approached using Hamiltonian structures similar to those in the underlying quantum dynamics. Such structures are particularly transparent in closed QHOs.…”