We present an efficient general method for realizing a quantum walk operator corresponding to an arbitrary sparse classical random walk. Our approach is based on Grover and Rudolph’s method for preparing coherent versions of efficiently integrable probability distributions [2]. This method is intended for use in quantum walk algorithms with polynomial speedups, whose complexity is usually measu...

A quantum master equation of the Lindblad form is obtained in this paper by considering the reduction of wave packets. Different classical equations are derived after exactly mapping such a quantum master equation to a continuous time random walk (CTRW). Although this CTRW is a quasiclassical walk, the effects due to the quantum interference can still be important in such a walk. It is shown th...

We present the first robust implementation of a coined quantum walk over five steps using only passive optical elements. By employing a fiber network loop we keep the amount of required resources constant as the walker's position Hilbert space is increased. We observed a non-Gaussian distribution of the walker's final position, thus characterizing a faster spread of the photon wave packet in co...

We derive a perturbation expansion for general interacting random walks, where steps are made on the basis of the history of the path. Examples of models where this expansion applies are reinforced random walk, excited random walk, the true (weakly) self-avoiding walk and loop-erased random walk. We use the expansion to prove a law of large numbers and central limit theorem for two models: (i) ...

We propose a new framework for turning quantum search algorithms that decide into quantum algorithms for finding a solution. Consider we are given an abstract quantum search algorithm A that can determine whether a target g exists or not. We give a general construction of another operator U that both determines and finds the target, whenever one exists. Our amplification method at most doubles ...

This note introduces a quantum random walk in R and proves the weak convergence of its rescaled n-step densities. Quantum walks of the type we consider in this note were introduced in [1], which defined and analyzed the Hadamard quantum walk on Z, and a “new type of convergence theorem” for such quantum walks on Z was discovered by Konno [4, 5]. A much simpler proof of Konno’s theorem has recen...

We consider discrete-time nearest-neighbor quantum walks on random environments in one dimension. Using the method based on a path counting, we present both quenched and annealed weak limit theorems for the quantum walk.

We study the localization of probability distribution in a discrete quantum random walk on an infinite chain. With a phase defect introduced in any position of the quantum random walk (QRW), we have found that the localization of the probability distribution in the QRW emerges. Different localized behaviors of the probability distribution in the QRW are presented when the defect occupies differ...