Quantum Annealing - Foundations and Frontiers

We briefly review various computational methods for the solution of optimization problems. First, several classical methods such as Metropolis algorithm and simulated annealing are discussed. We continue with a description of quantum methods, namely adiabatic quantum computation and quantum annealing. Next, the new D-Wave computer and the recent progress in the field claimed by the D-Wave group are discussed. We present a set of criteria which could help in testing the quantum features of these computers. We conclude with a list of considerations with regard to future research.Comment: 22 pages, 6 figures. EPJ-ST Discussion and Debate Issue: Quantum Annealing: The fastest route to large scale quantum computation?, Eds. A. Das, S. Suzuki (2014

An analog of Wolstenholme’s theorem

In this paper we shall prove an analogous version of Wolstenholme\u27s theorem, namely, given a prime number p>=2 and positive integers a,b,m such that p|-m, we shall determine the maximal prime power (p^e) which divides the numerator of the fraction (1/m=1/(m+p^b)+...+1/(m+(p^a-1)p^b), when written in reduced form, with the exception of one case, where p=2, b=1, m>1 and (2^a||m-1). In this exceptional case, a lower bound for e is given

An analog of Wolstenholme’s theorem

In this paper we shall prove an analogous version of Wolstenholme\u27s theorem, namely, given a prime number p>=2 and positive integers a,b,m such that p|-m, we shall determine the maximal prime power (p^e) which divides the numerator of the fraction (1/m=1/(m+p^b)+...+1/(m+(p^a-1)p^b), when written in reduced form, with the exception of one case, where p=2, b=1, m>1 and (2^a||m-1). In this exceptional case, a lower bound for e is given

A Holevo-Type Bound for a Hilbert Schmidt Distance Measure

We prove a new version of the Holevo bound employing the Hilbert-Schmidt norm instead of the Kullback-Leibler divergence. Suppose Alice is sending classical information to Bob using a quantum channel, while Bob is performing some projective measurement. We bound the classical mutual information in terms of the Hilbert-Schmidt norm by its quantum Hilbert-Schmidt counterpart. This constitutes a Holevo-type upper bound on the classical information transmission rate via a quantum channel. The resulting inequality is rather natural and intuitive relating classical and quantum expressions using the same measure.Comment: 8 pages. Accepted to Journal of Quantum Information Scienc

Introduction to Weak Measurements and Weak Values

We present a short review of the theory of weak measurement. This should serve as a map for the theory and an easy way to get familiar with the main results, problems and paradoxes raised by the theory.Quanta 2013; 2: 7–17