On Representations of Semigroups Having Hypercube-like Cayley Graphs

The $n-dimensional hypercube, or n-cube, is the Cayley graph of the Abelian group Z2n. A number of combinatorially-interesting groups and semigroups arise from modified hypercubes. The inherent combinatorial properties of these groups and semigroups make them useful in a number of contexts, including coding theory, graph theory, stochastic processes, and even quantum mechanics. In this paper, particular groups and semigroups whose Cayley graphs are generalizations of hypercubes are described, and their irreducible representations are characterized. Constructions of faithful representations are also presented for each semigroup. The associated semigroup algebras are realized within the context of Clifford algebras

Directionally-unbiased unitary optical devices in discrete-time quantum walks

The optical beam splitter is a widely-used device in photonics-based quantum information processing. Specifically, linear optical networks demand large numbers of beam splitters for unitary matrix realization. This requirement comes from the beam splitter property that a photon cannot go back out of the input ports, which we call “directionally-biased”. Because of this property, higher dimensional information processing tasks suffer from rapid device resource growth when beam splitters are used in a feed-forward manner. Directionally-unbiased linear-optical devices have been introduced recently to eliminate the directional bias, greatly reducing the numbers of required beam splitters when implementing complicated tasks. Analysis of some originally directional optical devices and basic principles of their conversion into directionally-unbiased systems form the base of this paper. Photonic quantum walk implementations are investigated as a main application of the use of directionally-unbiased systems. Several quantum walk procedures executed on graph networks constructed using directionally-unbiased nodes are discussed. A significant savings in hardware and other required resources when compared with traditional directionally-biased beam-splitter-based optical networks is demonstrated.Accepted manuscriptPublished versio

Nested Sampling Methods

Nested sampling (NS) computes parameter posterior distributions and makes Bayesian model comparison computationally feasible. Its strengths are the unsupervised navigation of complex, potentially multi-modal posteriors until a well-defined termination point. A systematic literature review of nested sampling algorithms and variants is presented. We focus on complete algorithms, including solutions to likelihood-restricted prior sampling, parallelisation, termination and diagnostics. The relation between number of live points, dimensionality and computational cost is studied for two complete algorithms. A new formulation of NS is presented, which casts the parameter space exploration as a search on a tree. Previously published ways of obtaining robust error estimates and dynamic variations of the number of live points are presented as special cases of this formulation. A new on-line diagnostic test is presented based on previous insertion rank order work. The survey of nested sampling methods concludes with outlooks for future research.Comment: Updated version incorporating constructive input from four(!) positive reports (two referees, assistant editor and editor). The open-source UltraNest package and astrostatistics tutorials can be found at https://johannesbuchner.github.io/UltraNest

The Impact of Randomisation in Load Balancing and Random Walks

The real world is full of uncertainties. Classical analyses usually favour deterministic cases, which in practice can be too restricted. Hence it motivates us to add in randomness to make models similar to practical situations. In this thesis, we mainly study two network problems taken from the distributed computing world: iterative load balancing and random walks. An interesting observation is that the problems we study, though not quite related regarding their real world applications, can be linked by the same mathematical toolkit: Markov chain theory. These problems have been heavily studied in the literature. However, their assumptions are mostly \emph{deterministic}, which causes less flexibility and generality to the real world settings. The novelty of this thesis is that we add randomness in these problems in order to observe worst cases vs. average cases (load balancing) and static cases vs. dynamic cases (random walks). For iterative load balancing, the randomness is added on the number of tasks over the entire network. Previous works often assumed worst case initial loads, which may be wasteful sometimes. Hence we relax this condition and assume the loads are drawn from different probability distributions. In particular, we no longer assume the initial loads are chosen by an adversary. Instead, we assume the initial loads on each processor are sampled from independent and identically distributed (i.i.d.) probability distributions. We then study the same problems as in classical settings, i.e., the time needed for the load balancing process to reach a sufficiently small discrepancy. Our main result implies that under such a regime, the time required to balance a network can be much faster. An insightful observation is that the load discrepancy is proportional to the term $t^{-1/4}$ where $t$ is the time used to run the protocol. This implies two main improvements compared with previous works: first, when the initial discrepancy is the same, our regime can reach small discrepancy faster; second, we have established a connection between the time and the discrepancy while previous analyses do not have. For random walks, the randomness is added on the network topologies. This means at each time step (considering discrete times), the underlying network can change randomly. In particular, we want the graph ``evolves'' instead of changing arbitrarily. To model the graph changing process, we adopt a model commonly used in the literature, i.e., the edge-Markovian model. If an edge does not exist between the two nodes, then it will appear in the next step with probability $p$, and if it does then in the next step it will disappear with probability $q$. This model can simulate real world scenarios such as adding friends with each other in social networks or a disruption between two remotely connected computers. Our main contributions regarding random walks include the following results. First, we divided the edge-Markovian graph model into different regimes in a parameterised way. This provides an intuitive path to similar analyses of dynamic graph models. Dynamic models are often hard to analyse in the field because of its complicated nature. We present a possible strategy to reach some feasible solutions by using parameters ($p, q$ above) to control the process. Second, we again analyse the random walk behaviours on such models. We have found that under certain regimes, the random walk still shows similar behaviours especially its mixing nature as in static settings. For the other regimes, we also show either weaker mixing or no mixing results

Exact sampling with Markov chains

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1996.Includes bibliographical references (p. 79-83).by David Bruce Wilson.Ph.D

Discrete-Time Quantum Walk - Dynamics and Applications

This dissertation presents investigations on dynamics of discrete-time quantum walk and some of its applications. Quantum walks has been exploited as an useful tool for quantum algorithms in quantum computing. Beyond quantum computational purposes, it has been used to explain and control the dynamics in various physical systems. In order to use the quantum walk to its fullest potential, it is important to know and optimize the properties purely due to quantum dynamics and in presence of noise. Various studies of its dynamics in the absence and presence of noise have been reported. We propose new approaches to optimize the dynamics, discuss symmetries and effect of noise on the quantum walk. Making use of its properties, we propose the use of quantum walk as an efficient new tool for various applications in physical systems and quantum information processing. In the first and second part of this dissertation, we discuss evolution process of the quantum walks, propose and demonstrate the optimization of discrete-time quantum walk using quantum coin operation from SU(2) group and discuss some of its properties. We investigate symmetry operations and environmental effects on dynamics of the walk on a line and an n-cycle highlighting the interplay between noise and topology. Using the properties and behavior of quantum walk discussed in part two, in part three we propose the application of quantum walk to realize quantum phase transition in optical lattice, that is to efficiently control and redistribute ultracold atoms in optical lattice. We also discuss the implementation scheme. Another application we consider is creation of spatial entanglement using quantum walk on a quantum many body system.Comment: 199 pages, 52 figures, Thesis completed during 2009 at University of Waterloo (IQC), V2 : Index of figures has been made compac

Continuous-time quantum computing

Quantum computation using continuous-time evolution under a natural hardware Hamiltonian is a promising near- and mid-term direction toward powerful quantum computing hardware. Continuous-time quantum computing (CTQC) encompasses continuous-time quantum walk computing (QW), adiabatic quantum computing (AQC), and quantum annealing (QA), as well as other strategies which contain elements of these three. While much of current quantum computing research focuses on the discrete-time gate model, which has an appealing similarity to the discrete logic of classical computation, the continuous nature of quantum information suggests that continuous-time quantum information processing is worth exploring. A versatile context for CTQC is the transverse Ising model, and this thesis will explore the application of Ising model CTQC to classical optimization problems. Classical optimization problems have industrial and scientific significance, including in logistics, scheduling, medicine, cryptography, hydrology and many other areas. Along with the fact that such problems often have straightforward, natural mappings onto the interactions of readily-available Ising model hardware makes classical optimization a fruitful target for CTQC algorithms. After introducing and explaining the CTQC framework in detail, in this thesis I will, through a combination of numerical, analytical, and experimental work, examine the performance of various forms of CTQC on a number of different optimization problems, and investigate the underlying physical mechanisms by which they operate.Open Acces

Percolation, statistical topography, and transport in random-media

A review of classical percolation theory is presented, with an emphasis on novel applications to statistical topography, turbulent diffusion, and heterogeneous media. Statistical topography involves the geometrical properties of the isosets (contour lines or surfaces) of a random potential psi(x). For rapidly decaying correlations of psi, the isopotentials fall into the same universality class as the perimeters of percolation clusters. The topography of long-range correlated potentials involves many length scales and is associated either with the correlated percolation problem or with Mandelbrot's fractional Brownian reliefs. In all cases, the concept of fractal dimension is particularly fruitful in characterizing the geometry of random fields. The physical applications of statistical topography include diffusion in random velocity fields, heat and particle transport in turbulent plasmas, quantum Hall effect, magnetoresistance in inhomogeneous conductors with the classical Hall effect, and many others where random isopotentials are relevant. A geometrical approach to studying transport in random media, which captures essential qualitative features of the described phenomena, is advocated.Physic