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IEEE Conference on Decision and Control (CDC)

A Gradient Descent approach to Quantum Equalization via Coherent LQG Control

Rebbecca TY Thien, Shanon L. Vuglar, Ian R. Petersen

Published Dec 16, 2024• 0 citations
Publisher

Abstract

This paper considers a modified gradient descent approach to the quantum equalization problem via coherent linear quadratic Gaussian (LQG) control for a class of linear quantum systems defined in terms of position and momentum operators. This problem is related to a recent result in formulating a quantum equalization problem. The primary focus of this paper is to find an optimal solution for the quantum equalization problem in the sense that it improves the cost function for the coherent LQG problem. An illustrative example is included which demonstrates the algorithm.

DOI
10.1109/cdc56724.2024.10886884

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We propose a method to design a suboptimal, coherent quantum LQG controller to solve a quantum equalization problem. Our method involves reformulating the problem as a control problem and then designing a classical LQG controller and implementing it as a quantum system. Illustrative examples are included which demonstrate the algorithm for both active and passive systems, i.e., systems where the dynamics are described in terms of both position and momentum operators and systems with dynamics in terms of annihilation operators only.
This paper is concerned with the Coherent Quantum Linear Quadratic Gaussian (CQLQG) control problem of finding a stabilizing measurement-free quantum controller for a quantum plant so as to minimize an infinite-horizon mean square performance index for the fully quantum closed-loop system. In comparison with the observation-actuation structure of classical controllers, the coherent quantum feedback is less invasive to the quantum dynamics and quantum information. Both the plant and the controller are open quantum systems whose dynamic 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, which are organised as quadratic constraints on the controller matrices and reflect the preservation of CCRs in time. The CQLQG problem is a constrained optimization problem for the steady-state quantum covariance matrix of the plant-controller system satisfying an algebraic Lyapunov equation. We propose a gradient descent algorithm equipped with adaptive stepsize selection for the numerical solution of the problem. The algorithm finds a local minimum of the LQG cost over the parameters of the Hamiltonian and coupling operators of a stabilizing PR quantum controller, thus taking the PR constraints into account. A convergence analysis of the proposed algorithm is presented.
This paper is concerned with designing linear quadratic Gaussian (LQG) controller based on coherent quantum observer. In more concrete terms, we aim to design a stabilizing coherent quantum controller, which possesses the property of a coherent quantum observer, for a linear quantum plant in order to minimize a mean-square cost for the closed-loop quantum feedback system. In contrast to previous established work, we give a numerical approach to solving this problem via simulated annealing algorithm. Moreover, an illustrative example of designing locally optimal coherent quantum LQG controller are provided in order to demonstrate the algorithm performance proposed in the paper.
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The paper is concerned with the coherent quantum Linear Quadratic Gaussian (CQLQG) control problem for time-varying quantum plants governed by linear quantum stochastic differential equations over a bounded time interval. A controller is sought among quantum linear systems satisfying physical realizability (PR) conditions. The latter describe the dynamic equivalence of the system to an open quantum harmonic oscillator and relate its state-space matrices to the free Hamiltonian, coupling and scattering operators of the oscillator. Using the Hamiltonian parameterization of PR controllers, the CQLQG problem is recast into an optimal control problem for a deterministic system governed by a differential Lyapunov equation. The state of this subsidiary system is the symmetric part of the quantum covariance matrix of the plant-controller state vector. The resulting covariance control problem is treated using dynamic programming and Pontryagin's minimum principle. The associated Hamilton-Jacobi-Bellman equation for the minimum cost function involves Frechet differentiation with respect to matrix-valued variables. The gain matrices of the CQLQG optimal controller are shown to satisfy a quasi-separation property as a weaker quantum counterpart of the filtering/control decomposition of classical LQG controllers.