Quantum State Tomography with Observable Commutation Graphs

The measurement of quantum states has been a widely studied problem ever since the discovery of quantum mechanics. In general, we can only measure a quantum state once as the measurement itself alters the state and, consequently, we lose information about the original state of the system in the process. Furthermore, this single measurement cannot uncover every detail about the system's state and thus, we get only a limited description of the system. However, there are physical processes, e.g., a quantum circuit, which can be expected to create the same state over and over again. This allows us to measure multiple identical copies of the same system in order to gain a fuller characterization of the state. This process of diagnosing a quantum state through measurements is known as quantum state tomography. However, even if we are able to create identical copies of the same system, it is often preferable to keep the number of needed copies as low as possible. In this thesis, we will propose a method of optimising the measurements in this regard. The full description of the state requires determining multiple different observables of the system. These observables can be measured from the same copy of the system only if they commute with each other. As the commutation relation is not transitive, it is often quite complicated to find the best way to match the observables with each other according to these commutation relations. This can be quite handily illustrated with graphs. Moreover, the best way to divide the observables into commuting sets can then be reduced to a well-known graph theoretical problem called graph colouring. Measuring the observables with acceptable accuracy also requires measuring each observable multiple times. This information can also be included in the graph colouring approach by using a generalisation called multicolouring. Our results show that this multicolouring approach can offer significant improvements in the number of needed copies when compared to some other known methods

On a Clique-Based Integer Programming Formulation of Vertex Colouring with Applications in Course Timetabling

Vertex colouring is a well-known problem in combinatorial optimisation, whose alternative integer programming formulations have recently attracted considerable attention. This paper briefly surveys seven known formulations of vertex colouring and introduces a formulation of vertex colouring using a suitable clique partition of the graph. This formulation is applicable in timetabling applications, where such a clique partition of the conflict graph is given implicitly. In contrast with some alternatives, the presented formulation can also be easily extended to accommodate complex performance indicators (``soft constraints'') imposed in a number of real-life course timetabling applications. Its performance depends on the quality of the clique partition, but encouraging empirical results for the Udine Course Timetabling problem are reported

Graph Colouring and Frequency Assignment

In this thesis we study some graph colouring problems which arise from mathematical models of frequency assignment in radiocommunications networks, in particular from models formulated by Hale and by Tesman in the 1980s. The main body of the thesis is divided into four chapters. Chapter 2 is the shortest, and is largely self-contained; it contains some early work on the frequency assignment problem, in which each edge of a graph is assigned a positive integer weight, and an assignment of integer colours to the vertices is sought in which the colours of adjacent vertices differ by at least the weight of the edge joining them. The remaining three chapters focus on problems which combine frequency assignment with list colouring, in which each vertex has a list of integers from which its colour must be chosen. In Chapter 3 we study list colourings where the colours of adjacent vertices must differ by at least a fixed integer s, and in Chapter 4 we add the additional restriction that the lists must be sets of consecutive integers. In both cases we investigate the required size of the lists so that a colouring can always be found. By considering the behaviour of these parameters as s tends to infinity, we formulate continuous analogues of the two problems, considering lists which are real intervals in Chapter 4, and arbitrary closed real sets in Chapter 5. This gives rise to two new graph invariants, the consecutive choosability ratio tau(G) and the choosability ratio sigma(G). We relate these to other known graph invariants, provide general bounds on their values, and determine specific values for various classes of graphs

Graph Colouring and Frequency Assignment

In this thesis we study some graph colouring problems which arise from mathematical models of frequency assignment in radiocommunications networks, in particular from models formulated by Hale and by Tesman in the 1980s. The main body of the thesis is divided into four chapters. Chapter 2 is the shortest, and is largely self-contained; it contains some early work on the frequency assignment problem, in which each edge of a graph is assigned a positive integer weight, and an assignment of integer colours to the vertices is sought in which the colours of adjacent vertices differ by at least the weight of the edge joining them. The remaining three chapters focus on problems which combine frequency assignment with list colouring, in which each vertex has a list of integers from which its colour must be chosen. In Chapter 3 we study list colourings where the colours of adjacent vertices must differ by at least a fixed integer s, and in Chapter 4 we add the additional restriction that the lists must be sets of consecutive integers. In both cases we investigate the required size of the lists so that a colouring can always be found. By considering the behaviour of these parameters as s tends to infinity, we formulate continuous analogues of the two problems, considering lists which are real intervals in Chapter 4, and arbitrary closed real sets in Chapter 5. This gives rise to two new graph invariants, the consecutive choosability ratio tau(G) and the choosability ratio sigma(G). We relate these to other known graph invariants, provide general bounds on their values, and determine specific values for various classes of graphs

Numerical simulation of non-Newtonian fluid flow in mixing geometries

In this thesis, a theoretical investigation is undertaken into fluid and mixing flows generated by various geometries for Newtonian and non-Newtonian fluids, on both sequential and parallel computer systems. The thesis begins by giving the necessary background to the mixing process and a summary of the fundamental characteristics of parallel architecture machines. This is followed by a literature review which covers accomplished work in mixing flows, numerical methods employed to simulate fluid mechanics problems and also a review of relevant parallel algorithms. Next, an overview is given of the numerical methods that have been reviewed, discussing the advantages and disadvantages of the different methods. In the first section of the work the implementation of the primitive variable finite element method to solve a simple two dimensional fluid flow problem is studied. For the same geometry colour band mixing is also investigated. Further investigational work is undertaken into the flows generated by various rotors for both Newtonian and non-Newtonian fluids. An extended version of the primitive variable formulation is employed, colour band mixing is also carried out on two of these geometries. The latter work is carried out on a parallel architecture machine. The design specifications of a parallel algorithm for a MIMD system are discussed, with particular emphasis placed on frontal and multifrontal methods. This is followed by an explanation of the implementation of the proposed parallel algorithm, applied to the same fluid flow problems as considered earlier and a discussion of the efficiency of the system is given. Finally, a discussion of the conclusions of the entire accomplished work is presented. A number of suggestions for future work are also given. Three published papers relating to the work carried out on the transputer networks are included in the appendices