Diego de Falco

AN INTRODUCTION TO QUANTUM ANNEALING

Quantum annealing, or quantum stochastic optimization, is a classical randomized algorithm which provides good heuristics for the solution of hard optimization problems. The algorithm, suggested by the behaviour of quantum systems, is an example of proficuous cross contamination between classical and quantum computer science. In this survey paper we illustrate how hard combinatorial problems are tackled by quantum computation and present some examples of the heuristics provided by quantum annealing. We also present preliminary results about the application of quantum dissipation (as an alternative to imaginary time evolution) to the task of driving a quantum system toward its state of lowest energy.

DNA fragment assembly using neural prediction techniques.

The paper describes an alternative approach to the fragment assembly problem. The key idea is to train a recurrent neural network to tracking the sequence of bases constituting a given fragment and to assign to a same cluster all the sequences which are well tracked by this network. We make use of a 3-layer Recurrent Perceptron and examine both edited sequences from a ftp site and artificial fragments from a common simulation software: the clusters we obtain exhibit interesting properties in terms of error filtering, stability and self consistency; we define as well, with a certain degree of approximation, a metric on the fragment set. The proposed assembly algorithm is susceptible to becoming an alternative method with the following properties: (i) high quality of the rebuilt genomic sequences, (ii) high parallelizability of the computing process with consequent drastic reduction of the running time.

Applications of white noise calculus to the computation of Feynman integrals

We apply white noise calculus to the computation, according to the rigorous definitions given by T. Hida and L. Streit, of Feynman path integrals. More precisely, we show how the Feynman propagator in a uniform magnetic field can be explicitly computed in terms of P. Levys stochastic area spanned by two-dimensional Brownian motion. By the same technique we also compute the propagator for a quadratic non local action of relevance in some approximate calculations of quantum motion in the field of randomly located scatterers.

Speed and entropy of an interacting continuous time quantum walk

We present some dynamic and entropic considerations about the evolution of a continuous time quantum walk implementing the clock of an autonomous machine. On a simple model, we study in quite explicit terms the Lindblad evolution of the clocked subsystem, relating the evolution of its entropy to the spreading of the wave packet of the clock. We explore possible ways of reducing the generation of entropy in the clocked subsystem, as it amounts to a deficit in the probability of finding the target state of the computation. We are thus led to examine the benefits of abandoning some classical prejudice about how a clocking mechanism should operate.

Frequency of symbol occurrences in bicomponent stochastic models

We give asymptotic estimates of the frequency of occurrences of a symbol in a random word generated by any bicomponent stochastic model. More precisely, we consider the random variable Yn, representing the number of occurrences of a given symbol in a word of length n generated at random; the stochastic model is defined by a rational formal series r having a linear representation with two primitive components. This model includes the case when r is the product or the sum of two primitive rational formal series. We obtain asymptotic evaluations for the mean value and the variance of Yn and its limit distribution.

Learning by asymmetric parallel boltzmann machines

We consider the Little, Shaw, Vasudevan model as a parallel asymmetric Boltzmann machine, in the sense that we extend to this model the entropic learning rule first studied by Ackley, Hinton, and Sejnowski in the case of a sequentially activated network with symmetric synaptic matrix. The resulting Hebbian learning rule for the parallel asymmetric model draws the signal for the updating of synaptic weights from time averages of the discrepancy between expected and actual transitions along the past history of the network. As we work without the hypothesis of symmetry of the weights, we can include in our analysis also feedforward networks, for which the entropic learning rule turns out to be complementary to the error backpropagation rule, in that it rewards the correct behavior instead of penalizing the wrong answers.

Learning by parallel Boltzmann machines

A parallel implementation of the Boltzmann machine in which each unit is updated independently of, but simultaneously with, the other units is studied. A transparent representation of the transition matrix and of the equilibrium distribution emphasizes the role, for the stochastic parallel evolution of the dynamical features of the underlying synchronous deterministic Hopfield model. As a consequence of this fact, the parallel Boltzmann machine explores an energy landscape quite different from the one of the sequential model. It is shown that it is, nevertheless, possible to derive, for the parallel model, a realistic learning rule having the same feature of locality as the well-known learning rule for the sequential Boltzmann machine proposed by D. Ackley et al. (1985). >

Asymmetric Boltzmann machines

We study asymmetric stochastic networks from two points of view: combinatorial optimization and learning algorithms based on relative entropy minimization. We show that there are non trivial classes of asymmetric networks which admit a Lyapunov function ℒ under deterministic parallel evolution and prove that the stochastic augmentation of such networks amounts to a stochastic search for global minima of ℒ. The problem of minimizing ℒ for a totally antisymmetric parallel network is shown to be associated to an NP-complete decision problem. The study of entropic learning for general asymmetric networks, performed in the non equilibrium, time dependent formalism, leads to a Hebbian rule based on time averages over the past history of the system. The general algorithm for asymmetric networks is tested on a feed-forward architecture.

ENTROPY GENERATION IN A MODEL OF REVERSIBLE COMPUTATION

We present a model in which, due to the quantum nature of the signals controlling the implementation time of successive unitary computational steps, physical irreversibility appears in the execution of a logically reversible computation. Mathematics Subject Classification. 81p68.

PRAM layout optimisation

Abstract We propose a procedure for designing the layout of a fixed fan-in neural network based on the pRAM model. Using minimisation of the relative entropy between environment and output probability distribution laws as a target, and Amaris learning rule as a strategy, we show that consideration of the conditional entropy of the output of one node, given the candidate nodes that provide its inputs, leads to a locally optimum choice of the connections. The procedure is computationally feasible when certain preference criteria are used to control the mutual relevance of the pRAM nodes. We give some simple numerical examples of this procedure.