The quantum walk is a unitary analogue to the discrete random walk, and its properties have been increasingly studied since the turn of the millennium. In comparison with the classical random walk, the quantum walk exhibits linear spreading and initial condition dependent asymmetries. As noted early on in the conjecture and subsequent calculation of absorption probabilities in the one dimensional Hadamard walk, the interaction of the quantum walk with an absorbing boundary is fundamentally divergent from classical case. Here, we will survey absorption probabilities for a more general collection of one dimensional quantum walks and extend the method to consider d-dimensional walks in the presence of d-1 dimensional absorbing walls. However, these results are concerned only with local behavior at the boundary in the form of absorption probabilities. The main results of this thesis are concerned with the global behavior of finite quantum walks, which can be described by linear spreading in the short term, modal phenomena in the mid term, and stable distributions in the exceedingly long term. These theorems will be rigorously proved in the one-dimensional case and extrapolated to higher dimensional quantum walks. To this end we introduce QWSim, a new and robust computational engine for displaying finite two dimensional quantum walks.

Models of quantum walks which admit continuous time and continuous spacetime limits have recently led to quantum simulation schemes for simulating fermions in relativistic and non relativistic regimes (Di Molfetta and Arrighi, 2020). This work continues the study of relationships between discrete time quantum walks (DTQW) and their ostensive continuum counterparts by developing a more general framework than was done in (Di Molfetta and Arrighi, 2020) to evaluate the continuous time limit of these discrete quantum systems. Under this framework, we prove two constructive theorems concerning which internal discrete transitions ("coins") admit nontrivial continuum limits in 1D+1. We additionally prove that the continuous space limit of the continuous time limit of the DTQW can only yield massless states which obey the Dirac equation. We also demonstrate that for general coins the continuous time limit of the DTQW can be identified with the canonical continuous time quantum walk (CTQW) when the coin is allowed to transition through the continuum limit process. Finally, we introduce the Plastic Quantum Walk, or a quantum walk which admits both continuous time and continuous spacetime limits and, as a novel result, we use our 1D+1 results to obtain necessary and sufficient conditions concerning which DTQWs admit plasticity in 2D+1, showing the resulting Hamiltonians. We consider coin operators as general 4 parameter unitary matrices, with parameters which are functions of the lattice step size 𝜖. This dependence on 𝜖 encapsulates all functions of 𝜖 for which a Taylor series expansion in 𝜖 is well defined, making our results very general.

This dissertation consists of three sections. In the first section, we discuss the generation of arbitrary two-qubit entangled states and present three generation methods. The first method is based on the interaction of an atom with classical and quantized cavity fields. The second method is based on the interaction of two coupled two-level atoms with a laser field. In the last method, we use two spin-1/2 systems which interact with a tuned radio frequency pulse. Using those methods we have generated two qubit arbitrary entangled states which is widely used in quantum computing and quantum information. In the second section, we discuss a possible experimental implementation of quantum walk which is based on the passage of an atom through a high-Q cavity. The chirality is determined by the atomic states and the displacement is characterized by the photon number inside the cavity. Our scheme makes quantum walk possible in a cavity QED system and the results could be widely used on quantum computer. In the last section, we investigate the properties of teleporting an arbitrary superposition of entangled Dicke states of any number of atoms (qubits) between two distant cavities. We also studied teleporting continuous variables of an optical field. Teleportation of Dicke states relies on adiabatic passage using multiatom dark states in each cavity and a conditional detection of photons leaking out of both cavities. In the continuous variables teleportation scheme we first reformulate the protocol of quantum teleportation of arbitrary input optical field states in the density matrix form, and established the relation between the P-function of the input and output states. We then present a condition involving squeeze parameter and detection efficiency under which the P-function of the output state becomes the Q function of the input state such that any nonclassical features in the input state will be eliminated in the teleported state. Based on the research in this section we have made it possible of arbitrary atomic Dicke states teleportation from one cavity to another, and this teleortation will play an essential role in quantum communication. Since quantum properties is so important in quantum communication, the condition we give in this section to distinguish classical and quantum teleportation is also important.

Quantum walks on graphs have been proven to be a useful tool in quantum algorithm construction for various problems. In this thesis we introduce a new type of continuous-time quantum walk on graphs called the quantum snake walk, the basis states of which are fixed-length paths (snakes) in the underlying graph. We first consider the quantum snake walk on the line. The analysis of the eigenvalues and the eigenvectors of the Hamiltonian governing the walk reveals that most states initially localized in a segment on the line always remain in that same segment. However, there are exponentially small (in the length of the snake) fraction of states which move on the line as wave packets with momentum inversely proportional to the length of the snake. Next we show how an algorithm based on the quantum snake walk might be able to solve an extended version of the glued trees problem which asks to find a path connecting both roots of the glued trees graph. No efficient quantum algorithm solving this problem is known yet. For that reason we consider a specific extension of the glued trees graph and analyze how the quantum snake walk behaves on it. In particular we show that the quantum snake walk on the infinite binary tree, restricted to certain superpositions, in many aspects is very similar to the quantum snake walk on the line. We also argue why the quantum snake walk, initialized in certain superpositions on one side of the glued trees graph, after certain amount of time is likely to be found on the other side of the graph. This seems to be crucial if we want our algorithm to work.

Quantum walks on graphs have been proven to be a useful tool in quantum algorithm construction for various problems. In this thesis we introduce a new type of continuous-time quantum walk on graphs called the quantum snake walk, the basis states of which are fixed-length paths (snakes) in the underlying graph. We first consider the quantum snake walk on the line. The analysis of the eigenvalues and the eigenvectors of the Hamiltonian governing the walk reveals that most states initially localized in a segment on the line always remain in that same segment. However, there are exponentially small (in the length of the snake) fraction of states which move on the line as wave packets with momentum inversely proportional to the length of the snake. Next we show how an algorithm based on the quantum snake walk might be able to solve an extended version of the glued trees problem which asks to find a path connecting both roots of the glued trees graph. No efficient quantum algorithm solving this problem is known yet. For that reason we consider a specific extension of the glued trees graph and analyze how the quantum snake walk behaves on it. In particular we show that the quantum snake walk on the infinite binary tree, restricted to certain superpositions, in many aspects is very similar to the quantum snake walk on the line. We also argue why the quantum snake walk, initialized in certain superpositions on one side of the glued trees graph, after certain amount of time is likely to be found on the other side of the graph. This seems to be crucial if we want our algorithm to work.

This thesis studies the transition matrix of a quantum walk on strongly regular graphs. It is proposed by Emms, Hancock, Severini and Wilson in 2006, that the spectrum of a matrix based on the amplitudes of walks in the quantum walk, distinguishes strongly regular graphs. We begin by finding the eigenvalues of matrices describing the quantum walk for regular graphs. We also show that if two graphs are isomorphic, then the corresponding matrices produced by the procedure of Emms et al. are cospectral. We then look at the entries of the cube of the transition matrix and find an expression for the matrices produced by the procedure of Emms et al. in terms of the adjacency matrix and incidence matrices of the graph.

This thesis studies the transition matrix of a quantum walk on strongly regular graphs. It is proposed by Emms, Hancock, Severini and Wilson in 2006, that the spectrum of a matrix based on the amplitudes of walks in the quantum walk, distinguishes strongly regular graphs. We begin by finding the eigenvalues of matrices describing the quantum walk for regular graphs. We also show that if two graphs are isomorphic, then the corresponding matrices produced by the procedure of Emms et al. are cospectral. We then look at the entries of the cube of the transition matrix and find an expression for the matrices produced by the procedure of Emms et al. in terms of the adjacency matrix and incidence matrices of the graph.

Orientador: Marcos César de Oliveira / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin / Made available in DSpace on 2018-08-30T18:19:25Z (GMT). No. of bitstreams: 1 DoriguelloDiniz_JoaoFernando_M.pdf: 5802868 bytes, checksum: 75d67be6662dcc05e681b956a5848b12 (MD5) Previous issue date: 2016 / Resumo: Passeios quânticos são a contrapartida quântica aos passeios aleatórios clássicos e têm um papel crucial no desenho de algoritmos quânticos eficientes que superam algoritmos clássicos. Durante os últimos anos, um grande esforço tem sido feito para implementar passeios quânticos experimentalmente, principalmente unidimensionais. Entretanto, implementações experimentais de passeios quânticos em duas dimensões ainda são escassas, e é justamente a partir de duas dimensões em que há algum ganho computacional. Este projeto parte desta necessidade para propor um sistema físico capaz de implementar um passeio quântico bidimensional: um qubit superconductor acoplado capacitivamente a um oscilador mecânico. Este sistema já foi estudado anteriormente e mostrou-se ser capaz de reproduzir um passeio quântico unidimensional. O projeto centrou-se então em generalizar este sistema acoplamento um segundo oscilador mecânico ao qubit e em demonstrar que o sistema assim obtido é capaz, teoricamente, de gerar um passeio quântico bidimensional. Foram conduzidas simulações da distribuição de probabilidade do sistema sem e com decoerência. Além disso, estudamos a recente relação entre mecânica quântica relativística e passeios quânticos; mais especificamente, revisamos a semelhança entre as equações de Klein-Gordon e Dirac e as equações de evolução do passeio quântico / Abstract: Quantum walks are the quantum counterpart to classic random walks and have a crucial role in the design of efficient quantum algorithms that overcome classic algorithms. During the last years a great effort has been dedicated to implement quantum walks experimentally. However, bi-dimensional quantum walk implementations are still scarce and it is precisely starting from two dimensions that there exists some computational gain. This dissertation is based on this necessity to propose a physical system capable of implementing a two-dimensional quantum walk: a superconducting qubit capacitively coupled to a mechanical oscillator. This system has already been studied in the literature and was shown to adequately generate a one-dimensional quantum walk. The dissertation was thus centred in generalizing this system by coupling a second mechanical resonator to the qubit and in demonstrating that this resulting system is theoretically capable of simulating a two-dimensional quantum walk. Simulations of the probability distribution with and without decoherence were conducted. Moreover, we studied the recent connection between relativistic quantum mechanics and quantum walks; more specifically, we revised the resemblance between the Klein-Gordon and Dirac equations and the quantum walk time evolution equations / Mestrado / Física / Mestre em Física / 2014/12617-5 / FAPESP

In recent years, there has been a significant development in high performance computing, driven by advances in hardware and software technology. The performance of the computers to the present has improved in accordance with Moore’s law, on the other hand, it seems to be reaching the limits in the near future. The quantum computers, which have the potential to greatly exceed the capabilities of the classical computers, have been the focus of intense researches. In the present study, we investigate the difference of the classical random walk and the quantum walk based on theoretical point of view and the implementation in the simulation, and seek the applicability of the quantum walk in the future. We provide the overview of the fundamental theory in the classical random walk and the quantum walk, and compare the differences of the features, based on the behaviors between the classical random walk and the quantum walk, and the probability distributions. Also, we implement the quantum walk using the Qiskit as the quantum simulator. The quantum circuit representing the quantum walk is mainly composed of the three parts, the coin operator, the shift operator, and the quantum measurement. The coin operator represent the coin flip in the quantum walk, where we use the Hadamard operator. The shift operator indicates the movement of the quantum walk according to the result of the coin operator. The quantum measurement is the process of extracting the quantum state of qubits. In one-dimensional quantum walk, we prepare four cases, as the difference of the number of qubits for the position from two to five qubits. In all cases, the successful implementation of the quantum walk has been seen with respect to the number of qubits and the difference of the initial state. We then extensively investigate the implementation of the two-dimensional quantum walk. In two-dimensional quantum walk, three cases are prepared in terms of the number of qubits for the position in each x and y coordinates, from two to four qubits. Although the complexity of the problem setting is much increased compared to the one-dimensional case, the success of the quantum walk implementation can be seen. We also see that the behavior of the quantum walk and the spread of the probability distribution strongly depends on the initial condition in terms of both the initial coin state and the initial position. The present study has shown the applicability of the quantum walk as the tool for solving the complex problems in a wide range of future applications. In concluding remarks, we offer conceivable perspectives and future prospects of the present study.

För närvarande utforskas quantum walk som en potentiell metod för att analysera medicinska bilder. Med inspiration från Gradys random walk-algoritm för bildbehandling har vi utvecklat en metod som bygger på de kvantmekaniska fördelar som quantum walk innehar för att detektera och segmentera medicinska bilder. Vidare har de segmenterade bilderna utvärderats utifrån klinisk relevans. Teoretiskt sett kan quantum walk-algoritmer erbjuda en mer effektiv metod för bildanalys inom medicin jämfört med traditionella metoder för bildsegmentering som exempelvis klassisk random walk, som inte bygger på kvantmekanik. Inom området finns omfattande potential för utveckling, och det är av yttersta vikt att fortsätta utforska och förbättra metoder. För närvarande kan det konstateras att det är en lång väg att vandra innan detta är något som kan appliceras i en klinisk miljö. / Currently, quantum walk is being explored as a potential method for analyzing medical images. Taking inspiration from Grady's random walk algorithm for image processing, we have developed an approach that leverages the quantum mechanical advantages inherent in quantum walk to detect and segment medical images. Furthermore, the segmented images have been evaluated in terms of clinical relevance. Theoretically, quantum walk algorithms have the potential to offer a more efficient method for medical image analysis compared to traditional methods of image segmentation, such as classical random walk, which do not rely on quantum mechanics. Within this field, there is significant potential for development, and it is of utmost importance to continue exploring and refining these methods. However, it should be noted that there is a long way to go before this becomes something that can be applied in a clinical environment.