Search via Quantum Walk

We propose a new method for designing quantum search algorithms for finding a "marked" element in the state space of a classical Markov chain. The algorithm is based on a quantum walk \'a la Szegedy (2004) that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator. This operator is then used in an amplitude amplification scheme. As a result we considerably expand the scope of the previous approaches of Ambainis (2004) and Szegedy (2004). Our algorithm combines the benefits of these approaches in terms of being able to find marked elements, incurring the smaller cost of the two, and being applicable to a larger class of Markov chains. In addition, it is conceptually simple and avoids some technical difficulties in the previous analyses of several algorithms based on quantum walk.Comment: 21 pages. Various modifications and improvements, especially in Section

A Note on the Art of Network Design Problems

In this study, we describe some Network Design Problems (NDPs) as well as the network flowbased improvement algorithm for neighbourhood search defined by cycles. The main part of the study is structured around the formulation of the expected duration of stay in the educational system as a NDP. The fundamental matrix of the absorbing Markov chain is employed in computing the expected duration of each flow in the system. We shall illustrate the new graphtheoretic formulation for the educational system using datasets from a university setting. The paper concludes with suggestions for future directions of research.Keywords: absorbing Markov chain; educational system; graph theory; network design

One-Counter Stochastic Games

We study the computational complexity of basic decision problems for one-counter simple stochastic games (OC-SSGs), under various objectives. OC-SSGs are 2-player turn-based stochastic games played on the transition graph of classic one-counter automata. We study primarily the termination objective, where the goal of one player is to maximize the probability of reaching counter value 0, while the other player wishes to avoid this. Partly motivated by the goal of understanding termination objectives, we also study certain "limit" and "long run average" reward objectives that are closely related to some well-studied objectives for stochastic games with rewards. Examples of problems we address include: does player 1 have a strategy to ensure that the counter eventually hits 0, i.e., terminates, almost surely, regardless of what player 2 does? Or that the liminf (or limsup) counter value equals infinity with a desired probability? Or that the long run average reward is >0 with desired probability? We show that the qualitative termination problem for OC-SSGs is in NP intersection coNP, and is in P-time for 1-player OC-SSGs, or equivalently for one-counter Markov Decision Processes (OC-MDPs). Moreover, we show that quantitative limit problems for OC-SSGs are in NP intersection coNP, and are in P-time for 1-player OC-MDPs. Both qualitative limit problems and qualitative termination problems for OC-SSGs are already at least as hard as Condon's quantitative decision problem for finite-state SSGs.Comment: 20 pages, 1 figure. This is a full version of a paper accepted for publication in proceedings of FSTTCS 201

Nearly reducible finite Markov chains: theory and algorithms

Finite Markov chains are probabilistic network models that are commonly used as representations of dynamical processes in the physical sciences, biological sciences, economics, and elsewhere. Markov chains that appear in realistic modelling tasks are frequently observed to be nearly reducible, incorporating a mixture of fast and slow processes that leads to ill-conditioning of the underlying matrix of probabilities for transitions between states. Hence, the wealth of established theoretical results that makes Markov chains attractive and convenient models often cannot be used straightforwardly in practice, owing to numerical instability associated with the standard computational procedures to evaluate the expressions. This work is concerned with the development of theory, algorithms, and simulation methods for the efficient and numerically stable analysis of finite Markov chains, with a primary focus on exact approaches that are robust and therefore applicable to nearly reducible networks. New methodologies are presented to determine representative paths, identify the dominant transition mechanisms for a particular process of interest, and analyze the local states that have a strong influence on the characteristics of the global dynamics. The novel approaches yield new insights into the behaviour of Markovian networks, addressing and overcoming numerical challenges. The methodology is applied to example models that are relevant to current problems in chemical physics, including Markov chains representing a protein folding transition, and a configurational transition in an atomic cluster. Relevant classical theory of finite Markov chains and a description of existing robust algorithms for their numerical analysis is given in Chapter 1. The remainder of this thesis considers the problem of investigating a transition from an initial set of states in a Markovian network to an absorbing (target) macrostate. A formal approach to determine a finite set of representative transition paths is proposed in Chapter 2, based on exact pathwise decomposition of the total productive flux. This analysis allows for the importance of competing dynamical processes to be rigorously quantified. A robust state reduction algorithm to compute the expectation of any path property for a transition between two endpoint states is also described in Chapter 2. Chapter 3 reports further numerically stable state reduction algorithms to compute quantities that characterize the features of a transition at a statewise level of detail, allowing for identification of the local states that play a key role in modulating the slow dynamics. An expression is derived for the probability that a state is visited on a path that proceeds directly to the absorbing state without revisiting the initial state, which characterizes the dynamical relevance of an individual state to the overall transition process. In Chapter 4, an unsupervised strategy is proposed to utilize a highly efficient simulation algorithm for sampling paths on a Markov chain. The framework employs a scalable community detection algorithm to obtain an initial clustering of the network into metastable sets of states, which is subsequently refined by a variational optimization procedure. The optimized clustering is then used as the basis for simulating trajectory segments that necessarily escape from the metastable macrostates. The thesis is concluded with an overview of recent related advances that are beyond the scope of the current work (Chapter 5), and a discussion of potential applications where the novel methodology reported herein may be applied to perform insightful analyses that were previously intractable.Cambridge Commonwealth, European and International Trust Engineering and Physical Sciences Research Counci

Free parafermions

The spectrum of the quantum Ising chain can be found by expressing the spins in terms of free fermions. An analogous transformation exists for clock chains with $Z_n$ symmetry, but is of less use because the resulting parafermionic operators remain interacting. Nonetheless, Baxter showed that a certain non-hermitian (but PT-symmetric) clock Hamiltonian is "free", in the sense that the entire spectrum is found in terms of independent energy levels, with the striking feature that there are $n$ possibilities for occupying each level. Here I show this directly explicitly finding shift operators obeying a $Z_n$ generalization of the Clifford algebra. I also find higher Hamiltonians that commute with Baxter's and prove their spectrum comes from the same set of energy levels. This thus provides an explicit notion of a "free parafermion". A byproduct is an elegant method for the solution of the Ising/Kitaev chain with spatially varying couplings.Comment: 44 pages. v2: minor rewriting, added several reference

Quantum Data Hiding

We expand on our work on Quantum Data Hiding -- hiding classical data among parties who are restricted to performing only local quantum operations and classical communication (LOCC). We review our scheme that hides one bit between two parties using Bell states, and we derive upper and lower bounds on the secrecy of the hiding scheme. We provide an explicit bound showing that multiple bits can be hidden bitwise with our scheme. We give a preparation of the hiding states as an efficient quantum computation that uses at most one ebit of entanglement. A candidate data hiding scheme that does not use entanglement is presented. We show how our scheme for quantum data hiding can be used in a conditionally secure quantum bit commitment scheme.Comment: 19 pages, IEEE style, 8 figures, submitted to IEEE Transactions on Information Theor

A tool for model-checking Markov chains

Markov chains are widely used in the context of the performance and reliability modeling of various systems. Model checking of such chains with respect to a given (branching) temporal logic formula has been proposed for both discrete [34, 10] and continuous time settings [7, 12]. In this paper, we describe a prototype model checker for discrete and continuous-time Markov chains, the Erlangen-Twente Markov Chain Checker EÎMC2, where properties are expressed in appropriate extensions of CTL. We illustrate the general benefits of this approach and discuss the structure of the tool. Furthermore, we report on successful applications of the tool to some examples, highlighting lessons learned during the development and application of EÎMC2

Multivariate phase type distributions - Applications and parameter estimation

Den bedst kendte univariate sandsynlighedsfordeling er normalfordelingen. Den er grundigt beskrevet i litteraturen inden for et bredt felt af anvendelsesområder. I de tilfælde, hvor det ikke er meningsfuldt at anvende normalfordelingen, findes alternative sandsynlighedsfordelinger som alle er godt beskrevet; mange af disse tilhører klassen af fasetypefordelinger. Fasetypefordelinger har adskillige fordele. De er alsidige forstået på den måde, at de kan benyttes til at tilnærme en vilkårlig sandsynlighedsfordeling defineret på den positive reelle akse. Der eksisterer generelle probabilistiske resultater for hele klassen af fasetypefordelinger, hvilket bidrager til anvendelsen af forskellige estimeringsmetoder på enten klassen af fasetypefordelinger eller dens delklasser. Disse egenskaber gør klassen af fasetypefordelinger til et interessant alternativ til normalfordelingen.Når det kommer til multivariate problemer, så er den multivariate normalfordeling den eneste generelle fordeling, der tillader parameterestimering og statistisk inferens. Desværre er kendskabet til egenskaberne af den multivariate fasetypefordeling stærk begrænset. Resultaterne for parameterestimering og inferensteori for den univariate fasetypefordeling indikerer et potentiale for lignende gode resultater for klassen af multivariate fasetypefordelinger. Mit ph.d.-studium var en del afWork Package 3 i UNITE-projektet. UNITEprojektet arbejder mod det overordnede mål at forbedre kvaliteten af beslutningsgrundlaget for projekter. Dette gøres ved at reducere systematisk model bias og ved at beskrive og reducere model usikkerheder generelt. Forskning har vist, at afvigelsen fra omkostningsestimater for infrastrukturprojekter tydeligvis ikke er normaltfordelt men i stedet hælder mod budgetoverskridelser. Denne skævhed kan beskrives med fasetypefordelinger. Cost-benefit-analyser bruges til at evaluere potentielle fremtidige projekter og til at udvikle pålidelige omkostningsvurderinger. Successiv Princippet er en gruppebaseret analysemetode, der primært bruges til at prædiktere omkostninger og varighed af mellem til store projekter. Vi mener, at den matematiske modellering, der ligger til grund for Successiv Princippet, kan forbedres. Vi foreslår derfor en ny tilgang til modellering af den samlede varighed af et projekt ved hjælp af univariate fasetypefordelinger. Den matematiske model er dernæst udvidet til også at beskrive korrelationen mellem projektvarighed og omkostninger nu baseret på bivariate fasetypefordelinger. Vores model kan anvendes til at forbedre estimater for varighed og omkostninger, og derved hjælpe projekters beslutningstagere til at træffe en optimal beslutning.Det arbejde, jeg har udført som en del af mit ph.d.-studium, sigtede efter at belyse klassen af multivariate fasetypefordelinger. Denne afhandling indeholder analytiske og numeriske resultater for parameterestimering og inferensteori for en gruppe af multivariate fasetypefordelinger. Resultaterne kan betragtes som et første skridt i retning af en mere tilbundsgående forståelse af multivariate fasetypefordelinger. Vi er imidlertid langt fra at have afdækket det fulde potentiale af generelle fasetypefordelinger. En dybere forståelse af multivariate fasetypefordelinger vil åbne op for et bredt felt af anvendelsesområder.Afhandlingen består af en opsummerende rapport og to videnskabelige artikler. Det bagvedliggende arbejde var udført i perioden 2010 til 2014.The best known univariate probability distribution is the normal distribution. It is used throughout the literature in a broad field of applications. In cases where it is not sensible to use the normal distribution alternative distributions are at hand and well understood, many of these belonging to the class of phase type distributions. Phase type distributions have several advantages. They are versatile in the sense that they can be used to approximate any given probability distribution on the positive reals. There exist general probabilistic results for the entire class of phase type distributions, allowing for different estimation methods for the whole class or subclasses of phase type distributions. These attributes make this class of distributions an interesting alternative to the normal distribution. When facing multivariate problems, the only general distribution that allows for estimation and statistical inference, is the multivariate normal distribution. Unfortunately only little is known about the general class of multivariate phase type distribution. Considering the results concerning parameter estimation and inference theory of univariate phase type distributions, the class of multivariate phase type distributions shows potential for similar great results.My PhD studies were part of the the work package 3 of the UNITE project. The overall goal of the UNITE project is to improve the decision support prior to deciding on a project by reducing systematic model bias and by quantifying and reducing model uncertainties.Research has shown that the errors on cost estimates for infrastructure projects clearly do not follow a normal distribution but is skewed towards cost overruns. This skewness can be described using phase type distributions. Cost benefit analysis assesses potential future projects and depend on reliable cost estimates. The Successive Principle is a group analysis method primarily used for analyzing medium to large projects in relation to cost or duration. We believe that the mathematical modeling used in the Successive Principle can be improved. We suggested a novel approach for modeling the total duration of a project using a univariate phase type distribution. The model is then extended to catch the correlation between duration and cost estimates using a bivariate phase type distribution. The use of our model can improve estimates for duration and costs and therefore help project management to make the optimal decisions. The work conducted during my PhD studies aimed at shedding light on the class of multivariate phase type distributions. This thesis contains analytical and numerical results for parameter estimations and inference theory for a family of multivariate phase type distributions. The results can be used as a stepping stone towards understanding multivariate phase type distributions better. However, we are far from uncovering the full potential of general multivariate phase type distributions. Deeper understanding of multivariate phase type distributions will open up a broad field of research areas they can be applied to.This thesis consists of a summary report and two research papers. The work was carried out in the period 2010 - 2014