Transition Probabilities in Generalized Quantum Search Hamiltonian Evolutions

A relevant problem in quantum computing concerns how fast a source state can be driven into a target state according to Schr\"odinger's quantum mechanical evolution specified by a suitable driving Hamiltonian. In this paper, we study in detail the computational aspects necessary to calculate the transition probability from a source state to a target state in a continuous time quantum search problem defined by a multi-parameter generalized time-independent Hamiltonian. In particular, quantifying the performance of a quantum search in terms of speed (minimum search time) and fidelity (maximum success probability), we consider a variety of special cases that emerge from the generalized Hamiltonian. In the context of optimal quantum search, we find it is possible to outperform, in terms of minimum search time, the well-known Farhi-Gutmann analog quantum search algorithm. In the context of nearly optimal quantum search, instead, we show it is possible to identify sub-optimal search algorithms capable of outperforming optimal search algorithms if only a sufficiently high success probability is sought. Finally, we briefly discuss the relevance of a tradeoff between speed and fidelity with emphasis on issues of both theoretical and practical importance to quantum information processing.Comment: 17 pages, 6 figures, 3 tables. Online ready in Int. J. Geometric Methods in Modern Physics (2020

Generalized Elliptical Distributions: Theory and Applications

The thesis recalls the traditional theory of elliptically symmetric distributions. Their basic properties are derived in detail and some important additional properties are mentioned. Further, the thesis concentrates on the dependence structures of elliptical or even meta-elliptical distributions using extreme value theory and copulas. Some recent results concerning regular variation and bivariate asymptotic dependence of elliptical distributions are presented. Further, the traditional class of elliptically symmetric distributions is extended to a new class of `generalized elliptical distributions' to allow for asymmetry. This is motivated by observations of financial data. All the ordinary components of elliptical distributions, i.e. the generating variate, the location vector and the dispersion matrix remain. Particularly, it is proved that skew-elliptical distributions belong to the class of generalized elliptical distributions. The basic properties of generalized elliptical distributions are derived and compared with those of elliptically symmetric distributions. It is shown that the essential properties of elliptical distributions hold also within the broader class of generalized elliptical distributions and some models are presented. Motivated by heavy tails and asymmetries observed in financial data the thesis aims at the construction of a robust dispersion matrix estimator in the context of generalized elliptical distributions. A `spectral density approach' is used for eliminating the generating variate. It is shown that the `spectral estimator' is an ML-estimator provided the location vector is known. Nevertheless, it is robust within the class of generalized elliptical distributions. The spectral estimator corresponds to an M-estimator developed 1983 by Tyler. But in contrast to the more general M-approach used by Tyler the spectral estimator is derived on the basis of classical maximum-likelihood theory. Hence, desired properties like, e.g., consistency, asymptotic efficiency and normality follow in a straightforward manner. Not only caused by the empirical evidence of extremes but also due to the inferential problems occuring for high-dimensional data the performance of the spectral estimator is investigated in the context of modern portfolio theory and principal components analysis. Further, methods of random matrix theory are discussed. They are suitable for analyzing high-dimensional covariance matrix estimates, i.e. given a small sample size relative to the number of dimensions. It is shown that the Marchenko-Pastur law fails if the sample covariance matrix is used in the context of elliptically of even generalized elliptically distributed and heavy tailed data. But substituting the sample covariance matrix by the spectral estimator resolves the problem and the classical arguments of random matrix theory remain valid

Linking discrete orthogonality with dilation and translation for incomplete sigma-pi neural networks of Hopfield-type

AbstractIn this paper, we show how to extend well-known discrete orthogonality results for complete sigma-pi neural networks on bipolar coded information in presence of dilation and translation of the signals. The approach leads to a whole family of functions being able to implement any given Boolean function. Unfortunately, the complexity of such complete higher order neural network realizations increases exponentially with the dimension of the signal space. Therefore, in practise one often only considers incomplete situations accepting that not all but hopefully the most relevant information or Boolean functions can be realized. At this point, the introduced dilation and translation parameters play an essential rôle because they can be tuned appropriately in order to fit the concrete representation problem as best as possible without any significant increase of complexity. In detail, we explain our approach in context of Hopfield-type neural networks including the presentation of a new learning algorithm for such generalized networks

Minimal redefinition of the OSV ensemble

In the interesting conjecture, Z_{BH} = |Z_{top}|^2, proposed by Ooguri, Strominger and Vafa (OSV), the black hole ensemble is a mixed ensemble and the resulting degeneracy of states, as obtained from the ensemble inverse-Laplace integration, suffers from prefactors which do not respect the electric-magnetic duality. One idea to overcome this deficiency, as claimed recently, is imposing nontrivial measures for the ensemble sum. We address this problem and upon a redefinition of the OSV ensemble whose variables are as numerous as the electric potentials, show that for restoring the symmetry no non-Euclidean measure is needful. In detail, we rewrite the OSV free energy as a function of new variables which are combinations of the electric-potentials and the black hole charges. Subsequently the Legendre transformation which bridges between the entropy and the black hole free energy in terms of these variables, points to a generalized ensemble. In this context, we will consider all the cases of relevance: small and large black holes, with or without D_6-brane charge. For the case of vanishing D_6-brane charge, the new ensemble is pure canonical and the electric-magnetic duality is restored exactly, leading to proper results for the black hole degeneracy of states. For more general cases, the construction still works well as far as the violation of the duality by the corresponding OSV result is restricted to a prefactor. In a concrete example we shall show that for black holes with non-vanishing D_6-brane charge, there are cases where the duality violation goes beyond this restriction, thus imposing non-trivial measures is incapable of restoring the duality. This observation signals for a deeper modification in the OSV proposal.Comment: 23 pages, v2: minor change

A Theory of Networks for Appxoimation and Learning

Learning an input-output mapping from a set of examples, of the type that many neural networks have been constructed to perform, can be regarded as synthesizing an approximation of a multi-dimensional function, that is solving the problem of hypersurface reconstruction. From this point of view, this form of learning is closely related to classical approximation techniques, such as generalized splines and regularization theory. This paper considers the problems of an exact representation and, in more detail, of the approximation of linear and nolinear mappings in terms of simpler functions of fewer variables. Kolmogorov's theorem concerning the representation of functions of several variables in terms of functions of one variable turns out to be almost irrelevant in the context of networks for learning. We develop a theoretical framework for approximation based on regularization techniques that leads to a class of three-layer networks that we call Generalized Radial Basis Functions (GRBF), since they are mathematically related to the well-known Radial Basis Functions, mainly used for strict interpolation tasks. GRBF networks are not only equivalent to generalized splines, but are also closely related to pattern recognition methods such as Parzen windows and potential functions and to several neural network algorithms, such as Kanerva's associative memory, backpropagation and Kohonen's topology preserving map. They also have an interesting interpretation in terms of prototypes that are synthesized and optimally combined during the learning stage. The paper introduces several extensions and applications of the technique and discusses intriguing analogies with neurobiological data

q-thermostatistics and the analytical treatment of the ideal Fermi gas

We discuss relevant aspects of the exact q-thermostatistical treatment for an ideal Fermi system. The grand canonical exact generalized partition function is given for arbitrary values of the nonextensivity index q, and the ensuing statistics is derived. Special attention is paid to the mean occupation numbers of single-particle levels. Limiting instances of interest are discussed in some detail, namely, the thermodynamic limit, considering in particular both the high- and low-temperature regimes, and the approximate results pertaining to the case q \sim 1 (the conventional Fermi-Dirac statistics corresponds to q=1). We compare our findings with previous Tsallis' literature.Comment: v2: comparison with conventional results and validity of approximations clarified, typos corrected; accepted for publication in Physica

On the Cut-Off Prescriptions Associated with Power-Law Generalized Thermostatistics

We revisit the cut-off prescriptions which are needed in order to specify completely the form of Tsallis' maximum entropy distributions. For values of the Tsallis entropic parameter $q>1$ we advance an alternative cut-off prescription and discuss some of its basic mathematical properties. As an illustration of the new cut-off prescription we consider in some detail the $q$-generalized quantum distributions which have recently been shown to reproduce various experimental results related to high $T_c$ superconductors

Coarse-grained distributions and superstatistics

We show an interesting connexion between the coarse-grained distribution function arising in the theory of violent relaxation for collisionless stellar systems (Lynden-Bell 1967) and the notion of superstatistics introduced recently by Beck & Cohen (2003). We also discuss the analogies and differences between the statistical equilibrium state of a multi-components self-gravitating system and the metaequilibrium state of a collisionless stellar system. Finally, we stress the important distinction between mixing entropies, generalized entropies, H-functions, generalized mixing entropies and relative entropies

The conjecturing process: perspectives in theory and implications in practice

[Abstract]: In this paper we analyze different types and stages of the conjecturing process. A classification of conjectures is discussed. A variety of problems that could lead to conjectures are considered from the didactical point of view. Results from a number of research studies are used to identify and investigate a number of questions related to the theoretical background of conjecturing as well as practical implications in the learning process