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A quantum walk is the quantum analogue of a random walk. While it is relatively well understood how quantum walks can speed up random walk hitting times, it is a long-standing open question to what extent quantum walks can speed up the…

We discuss the use of high-order quantum accelerator modes to achieve an atom optical realization of a biased quantum random walk. We first discuss how one can create co-existent quantum accelerator modes, and hence how momentum transfer…

We investigate the relation between transport properties and entanglement between the internal (spin) and external (position) degrees of freedom in one-dimensional discrete time quantum walks. We obtain closed-form expressions for the…

Quantum random walks are shown to have non-intuitive dynamics, which makes them an attractive area of study for devising quantum algorithms for well-known classical problems as well as those arising in the field of quantum computing. In…

Quantum walks with memory(QWM) are a type of modified quantum walks that record the walker's latest path. The general model of coined QWM is presented in Phys. Rev. A 93, 042323 (2016). In this paper, we present general model of Szegedy…

Quantum walk models have been used as an algorithmic tool for quantum computation and to describe various physical processes. This paper revisits the relationship between relativistic quantum mechanics and the quantum walks. We show the…

Quantum networks are complex systems formed by the interaction among quantum processors through quantum channels. Analogous to classical computer networks, quantum networks allow for the distribution of quantum computation among quantum…

A proof that continuous time quantum walks are universal for quantum computation, using unweighted graphs of low degree, has recently been presented by Childs [PRL 102 180501 (2009)]. We present a version based instead on the discrete time…

It was recently pointed out that identifiability of quantum random walks and hidden Markov processes underlie the same principles. This analogy immediately raises questions on the existence of hidden states also in quantum random walks and…

Coin and scattering are the two major formulations for discrete quantum walks models, each believed to have its own advantages in different applications. Although they are related in some cases, it was an open question their equivalence in…

Discrete time quantum walks are known to be universal for quantum computation. This has been proven by showing that they can simulate a universal gate set. In this paper we examine computation in terms of language acceptance and present two…

We extend the idea of a discrete-time quantum walk on a graph by placing a qubit on each vertex, and allowing the walker to interact with the qubit at its current position. We show that allowing for a controlled-Z interaction at each time…

This paper demonstrates and proves that the coordination of actions in a distributed swarm can be enhanced by using quantum entanglement. In particular, we focus on - Global and local simultaneous random walks, using entangled qubits that…

We consider quantum walks defined on arbitrary infinite graphs, parameterized by a family of scattering matrices attached to the vertices. Multiplying each scattering matrix by an i.i.d. random phase, we obtain a random scattering quantum…

Split-step quantum walks are models of supersymmetric quantum walk, and thus their Witten indices can be defined. We prove that the Witten index of a split-step quantum walk coincides with the difference between the winding numbers of…

We present an explicit formula for the characteristic polynomial of the transition matrix of the discrete-time quantum walk on a graph via the second weighted zeta function. As applications, we obtain new proofs for the results on spectra…

In this Chapter, we present some interesting properties of quantum walks on the line. We concentrate our attention in the emergence of invariance and provide some insights into the ultimate origin of the observed behavior. In the first part…

For a continuous-time quantum walk on a line the variance of the position observable grows quadratically in time, whereas, for its classical counterpart on the same graph, it exhibits a linear, diffusive, behaviour. A quantum walk, thus,…

Quantum walks provide simple models of various fundamental processes. It is pivotal to know when the dynamics underlying a walk lead to quantum advantages just by examining its statistics. A walk with many indistinguishable particles and…

We investigate the counterparts of random walk in universal quantum computing and their implementation using standard quantum circuits. Quantum walk have been recently well investigated for traversing graphs with certain oracles. We focus…