A discrete-time staggered quantum walk was recently introduced as a generalization that allows the unification of other versions, such as the coined and Szegedy's walk. However, it also produces forms of quantum walks not covered by previous versions. To explore their properties, we study here the staggered walk on a hexagonal lattice. Such a walk is defined using a set of overlapping tessellations that cover the graph edges, and each tessellation is a partition of the node set into cliques. The hexagonal lattice requires at least three tessellations. Each tessellation is associated with a local unitary operator and the product of the local operators defines the evolution operator of the staggered walk on the graph. After defining the evolution operator on the hexagonal lattice, we analyze the quantum walk dynamics with the focus on the position standard deviation and localization. We also obtain analytic results for the time complexity of spatial search algorithms with one marked node using cyclic boundary conditions.
Due to recent technological advances, quantum computing has gained notoriety. In this paradigm, the concept of quantum walk is fundamental for the development of algorithms for quantum computers. Among the existing quantum walk models, the staggered model, proposed by Portugal et al. stands out, since it includes the Szegedy model as a particular case, and part of the coined model. The staggered model uses the concept of tessellations on graphs to generate the quantum evolution operators that perform the walker's movements on the graph. Tessellations on graphs also have an interesting value for graph theory, since the parameter \textit{tessellation cover number}
T(G) is related to several other parameters present in the literature such as \textit{chromatic number}, \textit{chromatic index}, \textit{total chromatic number}, \textit{independent set number}, and \textit{clique number}. In addition, problems related to
T(G), such as \textsc{
t-tessellability}, \textsc{good tesellable recognition}, and \textsc{total good tessellable recognition} have deep relations with classic problems in graph theory involving graph coloring. In this work we present results in graph theory for the problems related to tessellations on graphs mentioned above, such as computational complexity, lower and upper bounds for
T(G), and the value of
T(G) for several graph classes. Furthermore, we present a quantum walk model based on \textit{total tessellation cover}, being pioneer in the use of vertices and edges as possible locations for the walker, simultaneously.
There are at least three models of discrete-time quantum walks (QWs) on graphs currently under active development. In this work we focus on the equivalence of two of them, known as Szegedy’s and staggered QWs. We give a formal definition of the staggered model and discuss generalized versions for searching marked vertices. Using this formal definition, we prove that any instance of Szegedy’s model is equivalent to an instance of the staggered model. On the other hand, we show that there are instances of the staggered model that cannot be cast into Szegedy’s framework. Our analysis also works when there are marked vertices. We show that Szegedy’s spatial search algorithms can be converted into search algorithms in staggered QWs. We take advantage of the similarity of those models to define the quantum hitting time in the staggered model and to describe a method to calculate the eigenvalues and eigenvectors of the evolution operator of staggered QWs. 1
The decoherence phenomenon has been widely studied in different types of quantum walks. In this work we show how to model decoherence inspired by percolation on staggered quantum walks. Two models of unitary noise are described: breaking polygons and breaking vertices. The evolution operators subject to these noises are obtained, and the equivalence to the coined quantum walk model is presented. Further, we numerically analyze the effect of these decoherence models on the two-dimensional grid of 4-cliques. We examine how these perturbations affect the quantum walk based search algorithm in this graph and how expanding the tessellation intersection can make it more robust against decoherence.
The interest in quantum walks has been steadily increasing during the last two decades. It is still worth to present new forms of quantum walks that might find practical applications and new physical behaviors. In this work, we define discrete-time quantum walks on arbitrary oriented graphs by partitioning a graph into tessellations, which is a collection of disjoint cliques that cover the vertex set. By using the adjacency matrices associated with the tessellations, we define local unitary operators, whose product is the evolution operator of our quantum walk model. We introduce a parameter, called alpha, that quantifies the amount of orientation. We show that the parameter alpha can be tuned in order to increase the amount of quantum walk-based transport on oriented graphs.
The staggered quantum walk is a model defined by the product of local operators associated with two or more independent graph tessellations. This model is versatile, encompasses several well-known discrete-time quantum walks, and inherits interesting features of the continuous-time quantum walk. We propose an implementation of the staggered quantum walk model with superconducting microwave resonators, where the required local operations are provided by the nearest neighbor interaction of the resonators coupled through superconducting quantum interference devices. The tunability of the interactions makes this system an excellent toolbox for this class of quantum walks. We focus on the one-dimensional case and discuss its generalization to a more general class known as triangle-free graphs.