arXiv subjects

Matthew R. James

Publications and source records attributed to Matthew R. James.

47 records · Page 3Linked to original sources

arXiv2013

A Popov Stability Condition for Uncertain Linear Quantum Systems

This paper considers a Popov type approach to the problem of robust stability for a class of uncertain linear quantum systems subject to unknown perturbations in the system Hamiltonian. A general stability result is given for a general class of perturbations to the system Hamiltonian. Then, the special case of a nominal linear quantum system is considered with quadratic perturbations to the system Hamiltonian. In this case, a robust stability condition is given in terms of a frequency domain condition which is of the same form as the standard Popov stability condition.

quant-ph↗

arXiv2012

On the response of quantum linear systems to single photon input fields

The purpose of this paper is to extend linear systems and signals theory to include single photon quantum signals. We provide detailed results describing how quantum linear systems respond to multichannel single photon quantum signals. In particular, we characterize the class of states (which we call {\em photon-Gaussian} states) that result when multichannel photons are input to a quantum linear system. We show that this class of quantum states is preserved by quantum linear systems. Multichannel photon-Gaussian states are defined via the action of certain creation and annihilation operators on Gaussian states. Our results show how the output states are determined from the input states through a pair of transfer function relations. We also provide equations from which output signal intensities can be computed. Examples from quantum optics are provided to illustrate the results.

quant-ph↗

arXiv2012

Robust Stability of Quantum Systems with a Nonlinear Coupling Operator

This paper considers the problem of robust stability for a class of uncertain quantum systems subject to unknown perturbations in the system coupling operator. A general stability result is given for a class of perturbations to the system coupling operator. Then, the special case of a nominal linear quantum system is considered with non-linear perturbations to the system coupling operator. In this case, a robust stability condition is given in terms of a scaled strict bounded real condition.

quant-ph↗

arXiv2012

Robust Stability of Uncertain Quantum Systems

This paper considers the problem of robust stability for a class of uncertain quantum systems subject to unknown perturbations in the system Hamiltonian. Some general stability results are given for different classes of perturbations to the system Hamiltonian. Then, the special case of a nominal linear quantum system is considered with either quadratic or non-quadratic perturbations to the system Hamiltonian. In this case, robust stability conditions are given in terms of strict bounded real conditions.

quant-ph↗

arXiv2011

Direct and Indirect Couplings in Coherent Feedback Control of Linear Quantum Systems

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 $H^\infty$ and LQG synthesis methods are extended to accommodate direct couplings using multistep optimization. Examples are given to illustrate the results.

quant-ph↗

arXiv2010

A reduced complexity numerical method for optimal gate synthesis

Although quantum computers have the potential to efficiently solve certain problems considered difficult by known classical approaches, the design of a quantum circuit remains computationally difficult. It is known that the optimal gate design problem is equivalent to the solution of an associated optimal control problem, the solution to which is also computationally intensive. Hence, in this article, we introduce the application of a class of numerical methods (termed the max-plus curse of dimensionality free techniques) that determine the optimal control thereby synthesizing the desired unitary gate. The application of this technique to quantum systems has a growth in complexity that depends on the cardinality of the control set approximation rather than the much larger growth with respect to spatial dimensions in approaches based on gridding of the space, used in previous literature. This technique is demonstrated by obtaining an approximate solution for the gate synthesis on $SU(4)$- a problem that is computationally intractable by grid based approaches.

math.OC↗

arXiv2010

Numerical Solution of the Dynamic Programming Equation for the Optimal Control of Quantum Spin Systems

The purpose of this paper is to describe the numerical solution of the Hamilton-Jacobi-Bellman (HJB) for an optimal control problem for quantum spin systems. This HJB equation is a first order nonlinear partial differential equation defined on a Lie group. We employ recent extensions of the theory of viscosity solutions from Euclidean space to Riemannian manifolds to interpret possibly non-differentiable solutions to this equation. Results from differential topology on the triangulation of manifolds are then used to develop a finite difference approximation method, which is shown to converge using viscosity solution techniques. An example is provided to illustrate the method.

quant-ph↗

arXiv2009

Quantum Dissipative Systems and Feedback Control Design by Interconnection

The purpose of this paper is to extend J.C. Willems' theory of dissipative systems to the quantum domain. This general theory, which combines perspectives from the quantum physics and control engineering communities, provides useful methods for analysis and design of dissipative quantum systems. We describe the interaction of the plant and a class of exosystems in general quantum feedback network terms. Our results include an infinitesimal characterization of the dissipation property, which generalizes the well-known Positive Real and Bounded Real Lemmas, and is used to study some properties of quantum dissipative systems. We also show how to formulate control design problems using quantum network models, which implements Willems' `control by interconnection' for open quantum systems. This control design formulation includes, for example, standard problems of stabilization, regulation, and robust control.

quant-ph↗

arXiv2008

Avoiding entanglement sudden-death via measurement feedback control in a quantum network

In this paper, we consider a linear quantum network composed of two distantly separated cavities that are connected via a one-way optical field. When one of the cavity is damped and the other is undamped, the overall cavity state obtains a large amount of entanglement in its quadratures. This entanglement however immediately decays and vanishes in a finite time. That is, entanglement sudden-death occurs. We show that the direct measurement feedback method proposed by Wiseman can avoid this entanglement sudden-death, and further, enhance the entanglement. It is also shown that the entangled state under feedback control is robust against signal loss in a realistic detector, indicating the reliability of the proposed direct feedback method in practical situations.

quant-ph↗

arXiv2008

Gate complexity using Dynamic Programming

The relationship between efficient quantum gate synthesis and control theory has been a topic of interest in the quantum control literature. Motivated by this work, we describe in the present article how the dynamic programming technique from optimal control may be used for the optimal synthesis of quantum circuits. We demonstrate simulation results on an example system on SU(2), to obtain plots related to the gate complexity and sample paths for different logic gates.

quant-ph↗

arXiv2006

A discrete invitation to quantum filtering and feedback control

The engineering and control of devices at the quantum-mechanical level--such as those consisting of small numbers of atoms and photons--is a delicate business. The fundamental uncertainty that is inherently present at this scale manifests itself in the unavoidable presence of noise, making this a novel field of application for stochastic estimation and control theory. In this expository paper we demonstrate estimation and feedback control of quantum mechanical systems in what is essentially a noncommutative version of the binomial model that is popular in mathematical finance. The model is extremely rich and allows a full development of the theory, while remaining completely within the setting of finite-dimensional Hilbert spaces (thus avoiding the technical complications of the continuous theory). We introduce discretized models of an atom in interaction with the electromagnetic field, obtain filtering equations for photon counting and homodyne detection, and solve a stochastic control problem using dynamic programming and Lyapunov function methods.

math.PR↗