Title:

Discrete quantum walks and quantum image processing

In this thesis we have focused on two topics: Discrete Quantum Walks and Quantum Image Processing. Our work is a contribution within the field of quantum computation from the perspective of a computer scientist. With the purpose of finding new techniques to develop quantum algorithms, there has been an increasing interest in studying Quantum Walks, the quantum counterparts of classical random walks. Our work in quantum walks begins with a critical and comprehensive assessment of those elements of classical random walks and discrete quantum walks on undirected graphs relevant to algorithm development. We propose a model of discrete quantum walks on an infinite line using pairs of quantum coins under different degrees of entanglement, as well as quantum walkers in different initial state configurations, including superpositions of corresponding basis states. We have found that the probability distributions of such quantum walks have particular forms which are different from the probability distributions of classical random walks. Also, our numerical results show that the symmetry properties of quantum walks with entangled coins have a nontrivial relationship with corresponding initial states and evolution operators. In addition, we have studied the properties of the entanglement generated between walkers, in a family of discrete Hadamard quantum walks on an infinite line with one coin and two walkers. We have found that there is indeed a relation between the amount of entanglement available in each step of the quantum walk and the symmetry of the initial coin state. However, as we show with our numerical simulations, such a relation is not straightforward and, in fact, it can be counterintuitive. Quantum Image Processing is a blend of two fields: quantum computation and image processing. Our aim has been to promote crossfertilisation and to explore how ideas from quantum computation could be used to develop image processing algorithms. Firstly, we propose methods for storing and retrieving images using nonentangled and entangled qubits. Secondly, we study a case in which 4 different values are randomly stored in a single qubit, and show that quantum mechanical properties can, in certain cases, allow better reproduction of original stored values compared with classical methods. Finally, we briefly note that entanglement may be used as a computational resource to perform hardwarebased pattern recognition of geometrical shapes that would otherwise require classical hardware and software.
