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The proceedings of the IEEE Conference on Control Applications/The proceedings of the ... IEEE Conference on Control Applications

Direct and indirect couplings in the interconnection of open two level quantum systems

Zibo Miao, M. R. James

Published Sep 1, 2015• 1 citation
Publisher

Abstract

Recently, the influences and applications of indirect and direct couplings in coherent feedback control of linear quantum stochastic systems have been investigated. However, bilinear quantum stochastic systems escape the realm of the current known results. Furthermore, as fast tunable coupling of a set of two level quantum systems (qubits) with high coherence plays a vital role in quantum computing operations, in this paper we study direct and indirect couplings in the interconnection of qubits. A physical model described by bilinear quantum stochastic differential equations is presented, and physical realizability conditions (necessary and sufficient algebraic constraints) are provided to guarantee these bilinear quantum stochastic differential equations to correspond to the dynamics of directly or indirectly coupled open two level systems. These results shed light on network synthesis of bilinear quantum stochastic systems and coherent control of multiple qubits.

DOI
10.1109/cca.2015.7320608

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Recently, the influences and applications of indirect and direct couplings in coherent feedback control of linear quantum stochastic systems have been investigated. However, bilinear quantum stochastic systems escape the realm of the current known results. Furthermore, as fast tunable coupling of a set of two level quantum systems (qubits) with high coherence plays a vital role in quantum computing operations, in this paper we study direct and indirect couplings in the interconnection of qubits. A physical model described by bilinear quantum stochastic differential equations is presented, and physical realizability conditions (necessary and sufficient algebraic constraints) are provided to guarantee these bilinear quantum stochastic differential equations to correspond to the dynamics of directly or indirectly coupled open two level systems. These results shed light on network synthesis of bilinear quantum stochastic systems and coherent control of multiple qubits.
Coherent feedback control considers purely quantum controllers in order to overcome disadvantages such as the acquisition of suitable quantum information, quantum error correction, etc. These approaches lack a systematic characterization of quantum realizability. Recently, a condition characterizing when a system described as a linear stochastic differential equation is quantum was developed. Such condition was named physical realizability, and it was developed for linear quantum systems satisfying the quantum harmonic oscillator canonical commutation relations. In this context, open two-level quantum systems escape the realm of the current known condition. When compared to linear quantum system, the challenges in obtaining such condition for such systems radicate in that the evolution equation is now a bilinear quantum stochastic differential equation and that the commutation relations for such systems are dependent on the system variables. The goal of this paper is to provide a necessary and sufficient condition for the preservation of the Pauli commutation relations, as well as to make explicit the relationship between this condition and physical realizability.
The purpose of this paper is to study and design direct and indirect couplings for use in coherent feedback control of a class of linear quantum stochastic systems. A general physical model for a nominal linear quantum system coupled directly and indirectly to external systems is presented. Fundamental properties of stability, dissipation, passivity, and gain for this class of linear quantum models are presented and characterized using complex Lyapunov equations and linear matrix inequalities (LMIs). Coherent H∞H^\inftyH∞ and LQG synthesis methods are extended to accommodate direct couplings using multistep optimization. Examples are given to illustrate the results.
The purpose of this paper is to study and design direct and indirect couplings for use in coherent feedback control of a class of linear quantum stochastic systems. A general physical model for a nominal linear quantum system coupled directly and indirectly to external systems is presented. Fundamental properties of stability, dissipation, passivity, and gain for this class of linear quantum models are presented and characterized using complex Lyapunov equations and linear matrix inequalities (LMIs). Coherent H∞and LQG synthesis methods are extended to accommodate direct couplings using multistep optimization. Examples are given to illustrate the results.
Recent theoretical and experimental investigations of coherent feedback quantum control, the feedback control of a quantum system with another quantum system, has raised the important problem of how to synthesize a class of quantum systems, called the class of linear quantum stochastic systems, from basic quantum optical components and devices in a systematic way. The synthesis theory sought in this case can be naturally viewed as a quantum analogue of linear electrical network synthesis theory and as such has potential for applications beyond the realization of coherent quantum feedback controllers. In earlier work, Nurdinet al.have established that an arbitrary linear quantum stochastic system can be realized as a cascade connection of simpler one degree of freedom quantum harmonic oscillators, together with a direct interaction Hamiltonian which is bilinear in the canonical operators of the oscillators. However, from an experimental perspective and based on current methods and technologies, direct interaction Hamiltonians are challenging to implement for systems with more than just a few degrees of freedom. In order to facilitate more tractable physical realizations of these systems, this technical note develops a new synthesis algorithm for linear quantum stochastic systems that relies solely on field-mediated interactions, including in implementation of the direct interaction Hamiltonian. Explicit synthesis examples are provided to illustrate the realization of two degrees of freedom linear quantum stochastic systems using the new algorithm.
This paper is concerned with coherent quantum control design for translation invariant networks of identical quantum stochastic systems subjected to external quantum noise. The network is modelled as an open quantum harmonic oscillator and is governed by a set of linear quantum stochastic differential equations. The dynamic variables of this quantum plant satisfy the canonical commutation relations. Similar large-scale systems can be found, for example, in quantum metamaterials and optical lattices. The problem under consideration is to design a stabilizing decentralized coherent quantum controller in the form of another translation invariant quantum system, directly coupled to the plant, so as to minimize a weighted mean square functional of the dynamic variables of the interconnected networks. We consider this problem in the thermodynamic limit of infinite network size and present first-order necessary conditions for optimality of the controller.
Quantum engineering has seen rapid growth in the past two decades. Physicists, mathematicians and engineers have been working in unison to control a number of diverse systems in the quantum regime. Quantum control involving feedback has become particularly topical, as using information gained from a system can lead to more stable operation of a control protocol. Quantum feedback can be split into two paradigms: measurement based feedback and coherent feedback. Although it is increasingly evident that retaining the coherence of the feedback signal provides an intrinsic advantage over measurement-based feedback, coherent feedback is still a new paradigm. In particular, there are a limited number of options for coherently estimating a state within a feedback loop. As a step towards better understanding and implementation of quantum feedback, this thesis reports on modelling, estimation and control of quantum systems. The first topic is physical realisability of a variety of quantum systems, especially finite level systems and their outgrowths. Bilinear quantum stochastic differential equations (QSDEs) have to be employed to characterise these systems, which is a significant complement to the previous work on linear QSDEs. In light of the fact that direct and indirect couplings play a vital role in quantum network and control, we also provide state-space models for different classes of coupled open quantum systems incorporating both bidirectional and directional interactions. The second topic is coherent observers and optimal filtering. It is well established that classically estimation using the Kalman filter can provide improved performance over direct feedback schemes, and similar demonstrations have been performed for measurement-based quantum feedback. In this thesis I present coherent observers, including least mean squares estimators, which are driven by the coherent output of a specified quantum plant and designed such that some subset of the observer and plant's expectation values converge in the asymptotic limit. Not only can estimators "observe" mean quantities of quantum plant, but also they can "estimate" quantum correlations such as entanglement. This is of much importance to coherent feedback design in the absence of measurement steps without loss of fidelity. Last but not least, several topics regarding stabilisation and control design of quantum systems are discussed in my thesis. To be specific, tools for quantum stability analysis (e.g. Lyapunov conditions) are provided. Both measurement-based feedback and coherent feedback control design are taken into account. A measurement-based optimal controller for opto-mechanical systems aimed at synchronising different mechanical modes is presented, which is application-oriented. Moreover, A general observer-based coherent feedback control framework is studied, and I demonstrate a pole-placement approach via coherent observers that can be applied to various scenarios.