We present a general framework for finding the time-optimal evolution and the optimal Hamiltonian for a quantum system with a given set of initial and final states. Our formulation is based on the variational principle and is analogous to that for the brachistochrone in classical mechanics. We reduce the problem to a formal equation for the Hamiltonian which depends on certain constraint functions specifying the range of available Hamiltonians. For some simple examples of the constraints, we explicitly find the optimal solutions.PACS numbers: 03.67.Lx, 03.65.Ca, 02.30.Xx, 02.30.Yy In quantum mechanics one can change a given state to another by applying a suitable Hamiltonian on the system. In many situations, e.g. quantum computation, it is desirable to know the pathway in the shortest time.In this paper we consider the problem of finding the time-optimal path for the evolution of a pure quantum state and the optimal driving Hamiltonian. Recently, a growing number of works related to this topic have appeared. For instance, Alvarez and Gómez [1] showed that the quantum state in Grover's algorithm [2], known as the optimal quantum search algorithm [3], actually follows a geodesic curve derived from the Fubini-Study metric in the projective space. Khaneja et al. [4] and Zhang et al.[5], using a Cartan decomposition scheme for unitary operations, discussed the time optimal way to realize a two-qubit universal unitary gate under the condition that one-qubit operations can be performed in an arbitrarily short time. On the other hand, Tanimura et al. [6] gave an adiabatic solution to the optimal control problem in holonomic quantum computation, in which a desired unitary gate is generated as the holonomy corresponding to the minimal length loop in the space of control parameters for the Hamiltonian. Schulte-Herbrüggen et al.[7] exploited the differential geometry of the projective unitary group to give the tightest known upper bounds on the actual time complexity of some basic modules of quantum algorithms. More recently, Nielsen [8] introduced a lower bound on the size of the quantum circuit necessary to realize a given unitary operator based on the geodesic distance, with a suitable metric, between the unitary and the identity operators. However, a general method for generating the time optimal Hamiltonian evolution of quantum states was not known until now.Here we are going to study this problem by exploiting the analogy with the so-called brachistochrone problem in classical mechanics and the elementary properties * Electronic address: carlini@th.phys.titech.ac.jp † Electronic address: ahosoya@th.phys.titech.ac.jp ‡ Electronic address: koike@phys.keio.ac.jp § Electronic address: okudaira@th.phys.titech.ac.jp of quantum mechanics. In ordinary quantum mechanics the initial state and the Hamiltonian of a physical system are given and one has to find the final state using the Schrödinger equation. In our work we generalize this framework so as to optimize a certain cost functional with respect to the Hamiltonian a...

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Extending our previous work on time optimal quantum state evolution [A. Carlini, A. Hosoya, T. Koike and Y. Okudaira, Phys. Rev. Lett. 96, 060503 (2006)], we formulate a variational principle for finding the time optimal realization of a target unitary operation, when the available Hamiltonians are subject to certain constraints dictated either by experimental or by theoretical conditions. Since the time optimal unitary evolutions do not depend on the input quantum state this is of more direct relevance to quantum computation. We explicitly illustrate our method by considering the case of a two-qubit system self-interacting via an anisotropic Heisenberg Hamiltonian and by deriving the time optimal unitary evolution for three examples of target quantum gates, namely the swap of qubits, the quantum Fourier transform and the entangler gate. We also briefly discuss the case in which certain unitary operations take negligible time.

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To probe naked spacetime singularities with waves rather than with particles we study the well posedness of initial value problems for test scalar fields with finite energy so that the natural function space of initial data is the Sobolev space. In the case of static and conformally static spacetimes we examine the essential selfadjointness of the time translation operator in the wave equation defined in the Hilbert space. For some spacetimes the classical singularity becomes regular if probed with waves while stronger classical singularities remain singular. If the spacetime is regular when probed with waves we may say that the spacetime is ''globally hyperbolic.'' ͓S0556-2821͑99͒00620-7͔

The effective evolution of an inhomogeneous cosmological model may be described in terms of spatially averaged variables. We point out that in this context, quite naturally, a measure arises which is identical to a fluid model of the 'Kullback-Leibler Relative Information Entropy', expressing the distinguishability of the local inhomogeneous mass density field from its spatial average on arbitrary compact domains. We discuss the time-evolution of 'effective information' and explore some implications. We conjecture that the information content of the Universe -measured by Relative Information Entropy of a cosmological model containing dust matter -is increasing.PACS numbers: 04.20.-q, 04.40.-b, 89.70.+c, 95.30.-k, 98.80., 98.80.Hw A MEASURE OF INHOMOGENEITY IN THE UNIVERSECosmology is based on the hypothesis of simplicity called the cosmological principle, i.e. homogeneity and isotropy. The departure of the actual mass distribution from the homogeneous universe model is quantified in terms of density contrast or a statistical quantity like the two-point correlation function, which both have been studied either by perturbation theory or numerical simulations. Behind these investigations there is a belief that the Universe is homogeneous on some large enough scale. This belief has to be quantitatively confronted with observation, explicitly introducing a measure of inhomogeneity for a domain of the Universe.In this Letter we propose a measure which quantifies the distinguishability of the actual mass distribution from its spatial average, borrowing a well-known concept in standard information theory. Suppose we are told that the probability distribution is {q i } and would like to examine how close this distribution is to the actual one {p i } by carrying out observations or coin tossing; the relevant quantity in information theory is the relative entropy, that this relative entropy is not symmetric for the two distributions {p i } and {q i }. It is known that this measure always decreases or stays the same under Markovian stochastic processes (i.e., a linear positive map). Namely, the actual distribution becomes less and less distinguishable from the priorly informed distribution due to the random process. In cosmology we are interested in how the real matter distribution is different from its spatial average. For a continuum the relevant quantity would bewhere ̺ is the actual distribution and · · · D its spatial average in the volume V D on the compact domain D of the manifold Σ. We shall conjecture that the measure S{̺|| ̺ Σ } continues to grow indefinitely, if Σ is compact. The resolution of the apparent discrepancy between the gravitational system and the ordinary stochastic system will be, (i) we are considering in cosmology a nonisolated system defined by a comoving region D in contrast to an isolated system for an ordinary stochastic process, and (ii) the time evolution dictated by Einstein's equations induces a negative feed-back due to the attractive nature of the gravitational force, which tends to make ...

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