A Tutorial on Clique Problems in Communications and Signal Processing

Since its first use by Euler on the problem of the seven bridges of K\"onigsberg, graph theory has shown excellent abilities in solving and unveiling the properties of multiple discrete optimization problems. The study of the structure of some integer programs reveals equivalence with graph theory problems making a large body of the literature readily available for solving and characterizing the complexity of these problems. This tutorial presents a framework for utilizing a particular graph theory problem, known as the clique problem, for solving communications and signal processing problems. In particular, the paper aims to illustrate the structural properties of integer programs that can be formulated as clique problems through multiple examples in communications and signal processing. To that end, the first part of the tutorial provides various optimal and heuristic solutions for the maximum clique, maximum weight clique, and $k$-clique problems. The tutorial, further, illustrates the use of the clique formulation through numerous contemporary examples in communications and signal processing, mainly in maximum access for non-orthogonal multiple access networks, throughput maximization using index and instantly decodable network coding, collision-free radio frequency identification networks, and resource allocation in cloud-radio access networks. Finally, the tutorial sheds light on the recent advances of such applications, and provides technical insights on ways of dealing with mixed discrete-continuous optimization problems

Decoherence on Staggered Quantum Walks

Decoherence phenomenon has been widely studied in different types of quantum walks. In this work we show how to model decoherence inspired by percolation on staggered quantum walks. Two models of unitary noise are described: breaking polygons and breaking vertices. The evolution operators subject to these noises are obtained and the equivalence to the coined quantum walk model is presented. Further, we numerically analyze the effect of these decoherence models on the two-dimensional grid of $4$-cliques. We examine how these perturbations affect the quantum walk based search algorithm in this graph and how expanding the tessellations intersection can make it more robust against decoherence.Comment: 17 pages, 14 figure

Quantum Computation, Markov Chains and Combinatorial Optimisation

This thesis addresses two questions related to the title, Quantum Computation, Markov Chains and Combinatorial Optimisation. The first question involves an algorithmic primitive of quantum computation, quantum walks on graphs, and its relation to Markov Chains. Quantum walks have been shown in certain cases to mix faster than their classical counterparts. Lifted Markov chains, consisting of a Markov chain on an extended state space which is projected back down to the original state space, also show considerable speedups in mixing time. We design a lifted Markov chain that in some sense simulates any quantum walk. Concretely, we construct a lifted Markov chain on a connected graph G with n vertices that mixes exactly to the average mixing distribution of a quantum walk on G. Moreover, the mixing time of this chain is the diameter of G. We then consider practical consequences of this result. In the second part of this thesis we address a classic unsolved problem in combinatorial optimisation, graph isomorphism. A theorem of Kozen states that two graphs on n vertices are isomorphic if and only if there is a clique of size n in the weak modular product of the two graphs. Furthermore, a straightforward corollary of this theorem and Lovász’s sandwich theorem is if the weak modular product between two graphs is perfect, then checking if the graphs are isomorphic is polynomial in n. We enumerate the necessary and sufficient conditions for the weak modular product of two simple graphs to be perfect. Interesting cases include complete multipartite graphs and disjoint unions of cliques. We find that all perfect weak modular products have factors that fall into classes of graphs for which testing isomorphism is already known to be polynomial in the number of vertices. Open questions and further research directions are discussed

Finding a marked node on any graph by continuous-time quantum walk

Spatial search by discrete-time quantum walk can find a marked node on any ergodic, reversible Markov chain $P$ quadratically faster than its classical counterpart, i.e.\ in a time that is in the square root of the hitting time of $P$. However, in the framework of continuous-time quantum walks, it was previously unknown whether such general speed-up is possible. In fact, in this framework, the widely used quantum algorithm by Childs and Goldstone fails to achieve such a speedup. Furthermore, it is not clear how to apply this algorithm for searching any Markov chain $P$. In this article, we aim to reconcile the apparent differences between the running times of spatial search algorithms in these two frameworks. We first present a modified version of the Childs and Goldstone algorithm which can search for a marked element for any ergodic, reversible $P$ by performing a quantum walk on its edges. Although this approach improves the algorithmic running time for several instances, it cannot provide a generic quadratic speedup for any $P$. Secondly, using the framework of interpolated Markov chains, we provide a new spatial search algorithm by continuous-time quantum walk which can find a marked node on any $P$ in the square root of the classical hitting time. In the scenario where multiple nodes are marked, the algorithmic running time scales as the square root of a quantity known as the extended hitting time. Our results establish a novel connection between discrete-time and continuous-time quantum walks and can be used to develop a number of Markov chain-based quantum algorithms.Comment: This version deals only with new algorithms for spatial search by continuous-time quantum walk (CTQW) on ergodic, reversible Markov chains. Please see arXiv:2004.12686 for results on the necessary and sufficient conditions for the optimality of the Childs and Goldstone algorithm for spatial search by CTQ