We present a heuristic algorithm for finding a graph H as a minor of a graph G that is practical for sparse G and H with hundreds of vertices. We also explain the practical importance of finding graph minors in mapping quadratic pseudo-boolean optimization problems onto an adiabatic quantum annealer.

We use difference sets to construct interesting sets of lines in complex space. Using (v,k,1)-difference sets, we obtain k^2-k+1 equiangular lines in C^k when k-1 is a prime power. Using semiregular relative difference sets with parameters (k,n,k,l) we construct sets of n+1 mutually unbiased bases in C^k. We show how to construct these difference sets from commutative semifields and that several known maximal sets of mutually unbiased bases can be obtained in this way, resolving a conjecture about the monomiality of maximal sets. We also relate mutually unbiased bases to spin models.Comment: 23 pages; no figures. Minor correction as pointed out in arxiv.org:1104.337

The current generation of D-Wave quantum annealing processor is designed to minimize the energy of an Ising spin configuration whose pairwise interactions lie on the edges of a Chimera graph C M,N,L . In order to solve an Ising spin problem with arbitrary pairwise interaction structure, the corresponding graph must be minor-embedded into a Chimera graph. We define a combinatorial class of native clique minors in Chimera graphs with vertex images of uniform, near minimal size, and provide a polynomial-time algorithm that finds a maximum native clique minor in a given induced subgraph of a Chimera graph. These minors allow improvement over recent work and have immediate practical applications in the field of quantum annealing.

This paper discusses techniques for solving discrete optimization problems using quantum annealing. Practical issues likely to affect the computation include precision limitations, finite temperature, bounded energy range, sparse connectivity, and small numbers of qubits. To address these concerns we propose a way of finding energy representations with large classical gaps between ground and first excited states, efficient algorithms for mapping non-compatible Ising models into the hardware, and the use of decomposition methods for problems that are too large to fit in hardware. We validate the approach by describing experiments with D-Wave quantum hardware for low density parity check decoding with up to 1000 variables.

We introduce the problem of constructing weighted complex projective 2-designs from the union of a family of orthonormal bases. If the weight remains constant across elements of the same basis, then such designs can be interpreted as generalizations of complete sets of mutually unbiased bases, being equivalent whenever the design is composed of d + 1 bases in dimension d. We show that, for the purpose of quantum state determination, these designs specify an optimal collection of orthogonal measurements. Using highly nonlinear functions on abelian groups, we construct explicit examples from d + 2 orthonormal bases whenever d + 1 is a prime power, covering dimensions d = 6, 10, and 12, for example, where no complete sets of mutually unbiased bases have thus far been found.

Attack tree (AT) is one of the widely used non‐state‐space models for security analysis. The basic formalism of AT does not take into account defense mechanisms. Defense trees (DTs) have been developed to investigate the effect of defense mechanisms using measures such as attack cost, security investment cost, return on attack (ROA), and return on investment (ROI). DT, however, places defense mechanisms only at the leaf nodes and the corresponding ROI/ROA analysis does not incorporate the probabilities of attack. In attack response tree (ART), attack and response are both captured but ART suffers from the problem of state‐space explosion, since solution of ART is obtained by means of a state‐space model. In this paper, we present a novel attack tree paradigm called attack countermeasure tree (ACT) which avoids the generation and solution of a state‐space model and takes into account attacks as well as countermeasures (in the form of detection and mitigation events). In ACT, detection and mitigation are allowed not just at the leaf node but also at the intermediate nodes while at the same time the state‐space explosion problem is avoided in its analysis. We study the consequences of incorporating countermeasures in the ACT using three case studies (ACT for BGP attack, ACT for a SCADA attack and ACT for malicious insider attacks). Copyright © 2011 John Wiley & Sons, Ltd.

A unitary design is a collection of unitary matrices that approximate the entire unitary group, much like a spherical design approximates the entire unit sphere. In this paper, we use irreducible representations of the unitary group to find a general lower bound on the size of a unitary t-design in U(d), for any d and t. We also introduce the notion of a unitary code -a subset of U(d) in which the trace inner product of any pair of matrices is restricted to only a small number of distinct values -and give an upper bound for the size of a code of degree s in U(d) for any d and s. These bounds can be strengthened when the particular inner product values that occur in the code or design are known. Finally, we describe some constructions of designs: we give an upper bound on the size of the smallest weighted unitary tdesign in U(d), and we catalogue some t-designs that arise from finite groups. *

The theoretical analysis of evolutionary algorithms is believed to be very important for understanding their internal search mechanism and thus to develop more efficient algorithms. This paper presents a simple mathematical analysis of the explorative search behavior of a recently developed metaheuristic algorithm called harmony search (HS). HS is a derivative-free real parameter optimization algorithm, and it draws inspiration from the musical improvisation process of searching for a perfect state of harmony. This paper analyzes the evolution of the population-variance over successive generations in HS and thereby draws some important conclusions regarding the explorative power of HS. A simple but very useful modification to the classical HS has been proposed in light of the mathematical analysis undertaken here. A comparison with the most recently published variants of HS and four other state-of-the-art optimization algorithms over 15 unconstrained and five constrained benchmark functions reflects the efficiency of the modified HS in terms of final accuracy, convergence speed, and robustness.