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.

Jan 1, 20171 citation

Quantum Random Walks

Swarnadeep Majumder, Austen Couvertier

Apr 27, 20181 citation

Conditional probability distributions of finite absorbing quantum walks

Parker Kuklinski

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 ${Q}_{n}$ , 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 $t=O({n}^{2})$ , and stability for $t=O({n}^{3})$ . At the end of this paper, we demonstrate the existence of fractional quantum revivals in other sufficiently regular quantum walk systems.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 ${Q}_{n}$, 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 $t=O({n}^{2})$, and stability for $t=O({n}^{3})$. At the end of this paper, we demonstrate the existence of fractional quantum revivals in other sufficiently regular quantum walk systems.

arXiv:1909.12680Physical Review AMar 6, 202011 citations

Absorption probabilities of discrete quantum mechanical systems

Parker Kuklinski

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.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.

Journal of Physics A Mathematical and TheoreticalAug 7, 20181 citation

Quantum random walks

Luigi Accardi, G. S. Watson

Lecture notes in mathematicsJan 1, 198923 citations

{Q}_{n}

, 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({n}^{2})

, and stability for

t=O({n}^{3})

. At the end of this paper, we demonstrate the existence of fractional quantum revivals in other sufficiently regular quantum walk systems.