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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…
One can view quantum mechanics as a generalization of classical probability theory that provides for pairwise interference among alternatives. Adopting this perspective, we ``quantize'' the classical random walk by finding, subject to a…
We study a particular class of trace-preserving completely positive maps, called PQ-channels, for which classical and quantum evolutions are isolated in a certain sense. By combining open quantum random walks with a notion of recurrence, we…
There are presently two models for quantum walks on graphs. The "coined" walk uses discrete time steps, and contains, besides the particle making the walk, a second quantum system, the coin, that determines the direction in which the…
The quantum walk (QW) is the term given to a family of algorithms governing the evolution of a discrete quantum system and as such has a founding role in the study of quantum computation. We contribute to the investigation of QW phenomena…
The iterated random walk is a random process in which a random walker moves on a one-dimensional random walk which is itself taking place on a one-dimensional random walk, and so on. This process is investigated in the continuum limit using…
Discrete-time quantum walks are considered a counterpart of random walks and the study for them has been getting attention since around 2000. In this paper, we focus on a quantum walk which generates a probability distribution splitting to…
A method for computing the Riesz $\alpha$-capacity, $0 < \alpha \le 2$, of a general set $K \subset \mathbb{R}^d$ is given. The method is based on simulations of isotropic $\alpha$-stable motion paths in $d$-dimensions. The familiar…
In this paper we analyze the behavior of quantum random walks. In particular we present several new results for the absorption probabilities in systems with both one and two absorbing walls for the one-dimensional case. We compute these…
We present a comprehensive classification of one-dimensional coined quantum walks on the infinite line, focusing on the spatial probability distributions they induce. Building on prior results, we identify all initial coin states that lead…
Under some mild condition, a random walk in the plane is recurrent. In particular each trajectory is dense, and a natural question is how much time one needs to approach a given small neighborhood of the origin. We address this question in…
The quantum walk is the quantum analogue of the well-known random walk, which forms the basis for models and applications in many realms of science. Its properties are markedly different from the classical counterpart and might lead to…
The discrete-time quantum walk (QW) is a quantum version of the random walk (RW) and has been widely investigated for the last two decades. Some remarkable properties of QW are well known. For example, QW has a ballistic spreading, i.e., QW…
This note introduces some examples of quantum random walks in d-dimensional Eucilidean space and proves the weak convergence of their rescaled n-step densities. One of the examples is called the Plancherel quantum walk because the "quantum…
Quantum walks and random walks bear similarities and divergences. One of the most remarkable disparities affects the probability of finding the particle at a given location: typically, almost a flat function in the first case and a…
A switching random walk, commonly known under the misnomer `oscillating random walk', is a real-valued Markov chain whose distribution of increments is determined by the sign of the current position. We explicitly identify an invariant…
The coined quantum walk is a discretization of the Dirac equation of relativistic quantum mechanics, and it is the basis of many quantum algorithms. We investigate how it searches the complete bipartite graph of $N$ vertices for one of $k$…
Quantum walks exhibit many unique characteristics compared to classical random walks. In the classical setting, self-avoiding random walks have been studied as a variation on the usual classical random walk. Classical self-avoiding random…
To what extent is the underlying distribution of a finitely supported unbiased random walk on $\mathbb{Z}$ determined by the sequence of times at which the walk returns to the origin? The main result of this paper is that, in various…
Quantum measurements play a fundamental role in quantum mechanics and quantum information processing, but it is not easy to implement generalized measurements, the most powerful measurements allowed by quantum mechanics. Here we propose a…