Optimal Control for Generating Quantum Gates in Open Dissipative Systems

Optimal control methods for implementing quantum modules with least amount of relaxative loss are devised to give best approximations to unitary gates under relaxation. The potential gain by optimal control using relaxation parameters against time-optimal control is explored and exemplified in numerical and in algebraic terms: it is the method of choice to govern quantum systems within subspaces of weak relaxation whenever the drift Hamiltonian would otherwise drive the system through fast decaying modes. In a standard model system generalising decoherence-free subspaces to more realistic scenarios, openGRAPE-derived controls realise a CNOT with fidelities beyond 95% instead of at most 15% for a standard Trotter expansion. As additional benefit it requires control fields orders of magnitude lower than the bang-bang decouplings in the latter.Comment: largely expanded version, superseedes v1: 10 pages, 5 figure

The Significance of the CC-Numerical Range and the Local CC-Numerical Range in Quantum Control and Quantum Information

This paper shows how C-numerical-range related new strucures may arise from practical problems in quantum control--and vice versa, how an understanding of these structures helps to tackle hot topics in quantum information. We start out with an overview on the role of C-numerical ranges in current research problems in quantum theory: the quantum mechanical task of maximising the projection of a point on the unitary orbit of an initial state onto a target state C relates to the C-numerical radius of A via maximising the trace function |\tr \{C^\dagger UAU^\dagger\}|. In quantum control of n qubits one may be interested (i) in having U\in SU(2^n) for the entire dynamics, or (ii) in restricting the dynamics to {\em local} operations on each qubit, i.e. to the n-fold tensor product SU(2)\otimes SU(2)\otimes >...\otimes SU(2). Interestingly, the latter then leads to a novel entity, the {\em local} C-numerical range W_{\rm loc}(C,A), whose intricate geometry is neither star-shaped nor simply connected in contrast to the conventional C-numerical range. This is shown in the accompanying paper (math-ph/0702005). We present novel applications of the C-numerical range in quantum control assisted by gradient flows on the local unitary group: (1) they serve as powerful tools for deciding whether a quantum interaction can be inverted in time (in a sense generalising Hahn's famous spin echo); (2) they allow for optimising witnesses of quantum entanglement. We conclude by relating the relative C-numerical range to problems of constrained quantum optimisation, for which we also give Lagrange-type gradient flow algorithms.Comment: update relating to math-ph/070200

Relative CC"-Numerical Ranges for Applications in Quantum Control and Quantum Information

Motivated by applications in quantum information and quantum control, a new type of $C$"-numerical range, the relative $C$"-numerical range denoted $W_K(C,A)$, is introduced. It arises upon replacing the unitary group U(N) in the definition of the classical $C$"-numerical range by any of its compact and connected subgroups $K \subset U(N)$. The geometric properties of the relative $C$"-numerical range are analysed in detail. Counterexamples prove its geometry is more intricate than in the classical case: e.g. $W_K(C,A)$ is neither star-shaped nor simply-connected. Yet, a well-known result on the rotational symmetry of the classical $C$"-numerical range extends to $W_K(C,A)$, as shown by a new approach based on Lie theory. Furthermore, we concentrate on the subgroup $SU_{\rm loc}(2^n) := SU(2)\otimes ... \otimes SU(2)$, i.e. the $n$-fold tensor product of SU(2), which is of particular interest in applications. In this case, sufficient conditions are derived for $W_{K}(C,A)$ being a circular disc centered at origin of the complex plane. Finally, the previous results are illustrated in detail for $SU(2) \otimes SU(2)$.Comment: accompanying paper to math-ph/070103

Incorporating basic needs to reconcile poverty and ecosystem services

This is the author accepted manuscript. The final version is available from Wiley via the DOI in this recordConservation managers frequently face the challenge of protecting and sustaining biodiversity without producing detrimental outcomes for (often poor) human populations that depend upon ecosystem services for their wellbeing. However, win-win solutions are often elusive and can mask trade-offs and negative outcomes for the wellbeing of particular groups of people. To deal with such trade-offs, approaches are needed to identify both ecological as well as social thresholds to determine the acceptable 'solution space' for conservation. Although human wellbeing as a concept has recently gained prominence among conservationists, they still lack tools to evaluate how their action affects human wellbeing in a given context. This paper presents the Theory of Human Needs in the context of conservation, building on an extensive historical application of needs approaches in international development. We detail an innovative participatory method, to evaluate how human needs are met, using locally relevant thresholds. We then establish the connections between human needs and ecosystem services. An application of this method in coastal East Africa identifies households who are in serious harm through not meeting different basic needs, and uncovers the role of ecosystem services in meeting these. Drawing from the international development and wellbeing literature, we suggest that this methodological approach, can help conservationists and planners balance poverty alleviation and biodiversity protection, ensure that conservation measures do not, at the very least, push individuals into serious harm and as a basis for monitoring the impacts of conservation on multidimensional poverty. This article is protected by copyright. All rights reserved.This paper results from the project Sustainable Poverty Alleviation from Coastal Ecosystem Services (SPACES) project number NE-K010484-1, funded by the Ecosystem Services for Poverty Alleviation (ESPA) programme. The ESPA programme is funded by the Department for International Development (DFID), the Economic and Social Research Council (ESRC), and the Natural Environment Research Council (NERC)

Optimal control of circuit quantum electrodynamics in one and two dimensions

Optimal control can be used to significantly improve multi-qubit gates in quantum information processing hardware architectures based on superconducting circuit quantum electrodynamics. We apply this approach not only to dispersive gates of two qubits inside a cavity, but, more generally, to architectures based on two-dimensional arrays of cavities and qubits. For high-fidelity gate operations, simultaneous evolutions of controls and couplings in the two coupling dimensions of cavity grids are shown to be significantly faster than conventional sequential implementations. Even under experimentally realistic conditions speedups by a factor of three can be gained. The methods immediately scale to large grids and indirect gates between arbitrary pairs of qubits on the grid. They are anticipated to be paradigmatic for 2D arrays and lattices of controllable qubits.Comment: Published version

Hamiltonian statistical mechanics

A framework for statistical-mechanical analysis of quantum Hamiltonians is introduced. The approach is based upon a gradient flow equation in the space of Hamiltonians such that the eigenvectors of the initial Hamiltonian evolve toward those of the reference Hamiltonian. The nonlinear double-bracket equation governing the flow is such that the eigenvalues of the initial Hamiltonian remain unperturbed. The space of Hamiltonians is foliated by compact invariant subspaces, which permits the construction of statistical distributions over the Hamiltonians. In two dimensions, an explicit dynamical model is introduced, wherein the density function on the space of Hamiltonians approaches an equilibrium state characterised by the canonical ensemble. This is used to compute quenched and annealed averages of quantum observables.Comment: 8 pages, 2 figures, references adde

Experimentally Realizable C-NOT Gate in a Flux Qubit/Resonator System

In this paper we present an experimentally realizable microwave pulse sequence that effects a Controlled NOT (C-NOT) gate operation on a Josephson junction-based flux-qubit/resonator system with high fidelity in the end state. We obtained a C-NOT gate process fidelity of 0.988 (0.980) for a two (three) qubit/resonator system under ideal conditions, and a fidelity of 0.903 for a two qubit/resonator system under the best, currently achieved, experimental conditions. In both cases, we found that "qubit leakage" to higher levels of the resonator causes a majority of the loss of fidelity, and that such leakage becomes more pronounced as decoherence effects increase.Comment: 7 pages, 4 figures, submitted to Phys. Rev.

Elasticity in ecosystem services: Exploring the variable relationship between ecosystems and human well-being

Although ecosystem services are increasingly recognized as benefits people obtain from nature, we still have a poor understanding of how they actually enhance multidimensional human well-being, and how well-being is affected by ecosystem change. We develop a concept of “ecosystem service elasticity” (ES elasticity) that describes the sensitivity of human well-being to changes in ecosystems. ES Elasticity is a result of complex social and ecological dynamics and is context dependent, individually variable, and likely to demonstrate nonlinear dynamics such as thresholds and hysteresis. We present a conceptual framework that unpacks the chain of causality from ecosystem stocks through flows, goods, value, and shares to contribute to the well-being of different people. This framework builds on previous conceptualizations, but places multidimensional well-being of different people as the final element. This ultimately disaggregated approach emphasizes how different people access benefits and how benefits match their needs or aspirations. Applying this framework to case studies of individual coastal ecosystem services in East Africa illustrates a wide range of social and ecological factors that can affect ES elasticity. For example, food web and habitat dynamics affect the sensitivity of different fisheries ecosystem services to ecological change. Meanwhile high cultural significance, or lack of alternatives enhance ES elasticity, while social mechanisms that prevent access can reduce elasticity. Mapping out how chains are interlinked illustrates how different types of value and the well-being of different people are linked to each other and to common ecological stocks. We suggest that examining chains for individual ecosystem services can suggest potential interventions aimed at poverty alleviation and sustainable ecosystems while mapping out of interlinkages between chains can help to identify possible ecosystem service trade-offs and winners and losers. We discuss conceptual and practical challenges of applying such a framework and conclude on its utility as a heuristic for structuring interdisciplinary analysis of ecosystem services and human wellbeing.This paper results from the project Sustainable Poverty Alleviation from Coastal Ecosystem Services (SPACES) project number NE-K010484-1, funded by the Ecosystem Services for Poverty Alleviation (ESPA) programme. The ESPA programme is funded by the Department for International Development (DFID), the Economic and Social Research Council (ESRC), and the Natural Environment Research Council (NERC).

Quantum Control at the Boundary

We present a scheme for controlling the state of a quantum system by modifying the boundary conditions. This constitutes an infinite-dimensional control problem. We provide conditions for the existence of solutions of the dynamics and prove that this system is approximately controllable