Network Synthesis of Linear Dynamical Quantum Stochastic Systems

The purpose of this paper is to develop a synthesis theory for linear dynamical quantum stochastic systems that are encountered in linear quantum optics and in phenomenological models of linear quantum circuits. In particular, such a theory will enable the systematic realization of coherent/fully quantum linear stochastic controllers for quantum control, amongst other potential applications. We show how general linear dynamical quantum stochastic systems can be constructed by assembling an appropriate interconnection of one degree of freedom open quantum harmonic oscillators and, in the quantum optics setting, discuss how such a network of oscillators can be approximately synthesized or implemented in a systematic way from some linear and non-linear quantum optical elements. An example is also provided to illustrate the theory.Comment: Revised and corrected version, published in SIAM Journal on Control and Optimization, 200

Mixed quantum-classical linear systems synthesis and quantum feedback control designs

This thesis makes some theoretical contributions towards mixed quantum feedback network synthesis, quantum optical realization of classical linear stochastic systems and quantum feedback control designs. A mixed quantum-classical feedback network is an interconnected system consisting of a quantum system and a classical system connected by interfaces that convert quantum signals to classical signal (using homodyne detectors), and vice versa (using electro-optic modulators). In the area of mixed quantum-classical feedback networks, we present a network synthesis theory, which provides a natural framework for analysis and design for mixed linear systems. Physical realizability conditions are derived for linear stochastic differential equations to ensure that mixed systems can correspond to physical systems. The mixed network synthesis theory developed based on physical realizability conditions shows that how a classical of mixed quantum-classical systems described by linear stochastic differential equations can be built as a interconnection of linear quantum systems and linear classical systems using quantum optical devices as well as electrical and electric devices. However, an important practical problem for the implementation of mixed quantum-classical systems is the relatively slow speed of classical parts implemented with standard electrical and electronic devices, since a mixed system will not work correctly unless the electronic processing of classical devices is fast enough. Therefore, another interesting work is to show how classical linear stochastic systems build using electrical and electric devices can be physically implemented using quantum optical components. A complete procedure is proposed for a stable quantum linear stochastic system realizing a given stable classical linear stochastic system. The thesis explains how it may be possible to realize certain measurement feedback loops fully at the quantum level. In the area of quantum feedback control design, two numerical procedures based on extended linear matrix inequality (LMI) approach are proposed to design a coherent quantum controller in this thesis. The extended synthesis linear matrix inequalities are, in addition to new analysis tools, less conservative in comparison to the conventional counterparts since the optimization variables related to the system parameters in extended LMIs are independent of the symmetric Lyapunov matrix. These features may be useful in the optimal design of quantum optical networks. Time delays are frequently encountered in linear quantum feedback control systems such as long transmission lines between quantum plants and linear controllers, which may have an effect on the performance of closed-loop plant controller systems. Therefore, this thesis investigates the problem of linear quantum measurement-based feedback control systems subject to feedback-loop time delay described by linear stochastic differential equations. Several numerical procedures are proposed to design classical controllers that make quantum measurement-based feedback control systems with time delay stable and also guarantee that their desired control performance specifications are satisfied

The SLH framework for modeling quantum input-output networks

Many emerging quantum technologies demand precise engineering and control over networks consisting of quantum mechanical degrees of freedom connected by propagating electromagnetic fields, or quantum input-output networks. Here we review recent progress in theory and experiment related to such quantum input-output networks, with a focus on the SLH framework, a powerful modeling framework for networked quantum systems that is naturally endowed with properties such as modularity and hierarchy. We begin by explaining the physical approximations required to represent any individual node of a network, eg. atoms in cavity or a mechanical oscillator, and its coupling to quantum fields by an operator triple $(S,L,H)$. Then we explain how these nodes can be composed into a network with arbitrary connectivity, including coherent feedback channels, using algebraic rules, and how to derive the dynamics of network components and output fields. The second part of the review discusses several extensions to the basic SLH framework that expand its modeling capabilities, and the prospects for modeling integrated implementations of quantum input-output networks. In addition to summarizing major results and recent literature, we discuss the potential applications and limitations of the SLH framework and quantum input-output networks, with the intention of providing context to a reader unfamiliar with the field.Comment: 60 pages, 14 figures. We are still interested in receiving correction

Quantum control theory and applications: A survey

This paper presents a survey on quantum control theory and applications from a control systems perspective. Some of the basic concepts and main developments (including open-loop control and closed-loop control) in quantum control theory are reviewed. In the area of open-loop quantum control, the paper surveys the notion of controllability for quantum systems and presents several control design strategies including optimal control, Lyapunov-based methodologies, variable structure control and quantum incoherent control. In the area of closed-loop quantum control, the paper reviews closed-loop learning control and several important issues related to quantum feedback control including quantum filtering, feedback stabilization, LQG control and robust quantum control.Comment: 38 pages, invited survey paper from a control systems perspective, some references are added, published versio