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.
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.
Quantum walks are known to have nontrivial interactions with absorbing boundaries. In particular it has been shown that an absorbing boundary in the one-dimensional quantum walk partially reflects information, as observed by absorption probability computations. In this paper, we shift our sights from the local phenomena of absorption probabilities to the global behavior of finite absorbing quantum walks in one dimension. We conduct our analysis by approximating the eigenbasis of the associated absorbing quantum walk operator matrix Qn, where n is the lattice size. The conditional probability distributions of these finite absorbing quantum walks exhibit distinct behavior at various timescales, namely wavelike reflections for times t=O(n), fractional quantum revivals for t=O(n2), and stability for t=O(n3). At the end of this paper, we demonstrate the existence of fractional quantum revivals in other sufficiently regular quantum walk systems.
Quantum walks are interesting models with potential applications to quantum algorithms and physical processes such as photosynthesis. In this paper, we study two models of one-dimensional quantum walks, namely, quantum walks with a moving absorbing wall and quantum walks with one stationary and one moving absorbing wall. For the former, we calculate numerically the survival probability, the rate of change of average position, and the rate of change of standard deviation of the particle's position in the long time limit for different wall velocities. Moreover, we also study the asymptotic behavior and the dependence of the survival probability on the initial particle's state. While for the latter, we compute the absorption probability of the right stationary wall for different velocities and initial positions of the left wall boundary. The results for these two models are compared with those obtained for the classical model. The difference between the results obtained for the quantum and classical models can be attributed to the difference in the probability distributions.
Absorbing boundaries introduce non-unitary dynamics in quantum mechanical systems. In particular, it has been shown that absorbing boundaries partially reflect quantum information in a variety of one-dimensional quantum walks. The methods used to compute absorption probabilities for quantum walks can be extended to calculate absorption probabilities for general finite and discrete quantum mechanical systems. These methods can also be used to compute moments of the associated absorption time distributions. We conclude the paper by discussing a modification of the quantum Zeno effect in these absorption systems.
In this paper, we present a simulator for two-particle quantum walks on the line and one-particle on a two-dimensional squared lattice. It can be used to investigate the equivalence between the two cases (one- and two-particle walks) for various boundary conditions (open, circular, reflecting, absorbing and their combinations). For the case of a single walker on a two-dimensional lattice, the simulator can also implement the Möbius strip. Furthermore, other topologies for the walker are also simulated by the proposed tool, like certain types of planar graphs with degree up to 4, by considering missing links over the lattice. The main purpose of the simulator is to study the genuinely quantum effects on the global properties of the two-particle joint probability distribution on the entanglement between the walkers/axis. For that purpose, the simulator is designed to compute various quantities such as: the entanglement and classical correlations, (classical and quantum) mutual information, the average distance between the two walkers, different hitting times and quantum discord. These quantities are of vital importance in designing possible algorithmic applications of quantum walks, namely in search, 3-SAT problems, etc. The simulator can also implement the static partial measurements of particle(s) positions and dynamic breaking of the links between certain nodes, both of which can be used to investigate the effects of decoherence on the walker(s). Finally, the simulator can be used to investigate the dynamic Anderson-like particle localization by varying the coin operators of certain nodes on the line/lattice. We also present some illustrative and relevant examples of one- and two-particle quantum walks in various scenarios. The tool was implemented in C and is available on-line at http://qwsim.weebly.com/ .
Quantum computing aims to harness and exploit the quantum mechanical phenomena such as superposition, entanglement and contextuality in order to encode and process information. In this context, quantum walks, which has been suggested as the quantum counterpart of classical random walks, is an emerging topic in quantum computing that provides powerful techniques for developing new quantum algorithms, quantum simulation and quantum state transfer. This thesis intends to investigate the properties of quantum walks which may potentially promote further work in such techniques in quantum computation. We first propose a novel method for transferring arbitrary unknown qubit state between two points in space with quantum walk architecture. We determine the cases providing perfect state transfers over both finite and infinite lattices with different boundary conditions and we introduce recovery operators assisting the transfer process. Next, by modeling the incoherent and coherent transport with classical random walks and quantum walks, respectively, we calculate the transport efficiencies over an explosive percolation lattice. We show that the minimal correlation between discrete clusters leads to maximal localizations which originating from random scatterings. These localization effects, however, are rather small when compared to the supportive effect of the abrupt growth of the largest cluster on transport efficiency, which eventually allows us to obtain more efficient transports in case of minimal correlations. We support our results with further calculations on whether the eigenstates of the systems we study are localized or not. Lastly, we turn our attention to the spreading dynamics and the coin-position entanglement for two-dimensional quantum walks under an artificial magnetic fields by introducing Peierls phases to the system. In particular, we show that the spreading of the quantum walk is diffusive rather than ballistic when the ratio of the magnetic flux through unit cell - where the walk takes place - to the flux quantum is an irrational number. On the contrary, the walk recovers its original ballistic behavior when this ratio is chosen to be a rational number. Furthermore, we demonstrate that coin-position entanglement is nearly maximum under an artificial magnetic field, even for large number of steps.
This manuscript gathers and subsumes a long series of works on using QW to simulate transport phenomena. Quantum Walks (QWs) consist of single and isolated quantum systems, evolving in discrete or continuous time steps according to a causal, shift-invariant unitary evolution in discrete space. We start reminding some necessary fundamentals of linear algebra, including the definitions of Hilbert space, tensor state, the definition of linear operator and then we briefly present the principles of quantum mechanics on which this thesis is grounded. After having reviewed the literature of QWs and the main historical approaches to their study, we then move on to consider a new property of QWs, the plasticity. Plastic QWs are those ones admitting both continuous time-discrete space and continuous spacetime time limit. We show that such QWs can be used to quantum simulate a large class of physical phenomena described by transport equations. We investigate this new family of QWs in one and two spatial dimensions, showing that in two dimensions, the PDEs we can simulate are more general and include dispersive terms. We show that the above results do not need to rely on the grid and we prove that such QW-based quantum simulators can be defined on 2-complex simplicia, i.e. triangular lattices. Finally, we extend the above result to any arbitrary triangulation, proving that such QWs coincide in the continuous limit to a transport equation on a general curved surface, including the curved Dirac equation in 2+1 spacetime dimensions.