Physical Realizability and Coherent LQG Control of Linear Quantum Systems

Abstract

Linear quantum systems are a special class of quantum systems whose dynamics are described by the laws of quantum mechanics. Quantum mechanics serves as a platform for comprehending and explaining the workings of the universe at the atomic scale. Control problems in the quantum domain are often more challenging compared to their classical counterparts, primarily due to the additional constraints imposed by quantum mechanics. A linear quantum system generally need not correspond to a physically meaningful system unless it satisfies some additional constraints in order to be a physically realizable quantum system. One way to implement a linear time-invariant~(LTI) system as a physically realizable quantum system is to include additional quantum vacuum noise channels. The presence of quantum vacuum noise channels in a quantum controller places limits on its performance. Hence it is desirable to minimize the number~(or effect) of these noises.

The first part of this thesis is to improve current approaches for implementing physically realizable quantum systems. In this context, we present an optimal method to implement a strictly proper, LTI system as a physically realizable quantum system. This method focuses on the extent to which the additional quantum noise affects the system output. We also give a necessary and sufficient condition for when a quantum system corresponding to a given LTI controller can be made physically realizable in the presence of both direct feedthrough quantum vacuum noise and additional quantum vacuum noise such that the additional quantum noise does not affect the controller output. Additionally, we give a frequency domain condition to physically realize a given transfer function matrix using only direct feedthrough quantum vacuum noise.

Coherent quantum control is a unique feedback control paradigm with no counterpart in classical control systems. Physical realizability and coherent quantum control are closely related concepts since the condition for a quantum controller to be considered coherent is that the controller must be physically realizable. The second part of this thesis considers the quantum equalization problem. We have proposed a method to find a physically realizable suboptimal coherent linear quadratic Gaussian~(LQG) controller that minimizes a cost function related to the system equalization error. Subsequently, we have implemented a gradient descent approach in searching for an optimal solution to the quantum equalization problem.