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

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.

Control aspects of quantum computing using pure and mixed states

Steering quantum dynamics such that the target states solve classically hard problems is paramount to quantum simulation and computation. And beyond, quantum control is also essential to pave the way to quantum technologies. Here, important control techniques are reviewed and presented in a unified frame covering quantum computational gate synthesis and spectroscopic state transfer alike. We emphasize that it does not matter whether the quantum states of interest are pure or not. While pure states underly the design of quantum circuits, ensemble mixtures of quantum states can be exploited in a more recent class of algorithms: it is illustrated by characterizing the Jones polynomial in order to distinguish between different (classes of) knots. Further applications include Josephson elements, cavity grids, ion traps and nitrogen vacancy centres in scenarios of closed as well as open quantum systems