Quantum walks (QWs) describe particles evolving coherently on a graph. The internal degree of freedom corresponds to a Hilbert space, called a coin system. We consider QWs on Cayley graphs of some group
G. In the literature, investigations concerning infinite
G have been focused on graphs corresponding to
G=Zd with a coin system of dimension 2, whereas for a one-dimensional coin (so-called scalar QWs) only the case of finite
G has been studied. Here we prove that the evolution of a scalar QW with
G infinite Abelian is trivial, providing a thorough classification of this kind of walks. Then we consider the infinite dihedral group ${D}_{\ensuremath{\infty}}$, that is, the unique non-Abelian group
G containing a subgroup $H\ensuremath{\cong}\mathbb{Z}$ with two cosets. We characterize the class of QWs on the Cayley graphs of ${D}_{\ensuremath{\infty}}$, and, via a coarse-graining technique, we show that it coincides with the class of spinorial walks on
Z which satisfies parity symmetry. This class of QWs includes the Weyl and the Dirac QWs. Remarkably, there exist also spinorial walks that are not coarse graining of a scalar QW, such as the Hadamard walk.
We show that certain types of quantum walks can be modeled as waves that propagate in a medium with phase and group velocities that are explicitly calculable. Since the group and phase velocities indicate how fast wave packets can propagate causally, we propose the use of these wave velocities in our definition for the hitting time of quantum walks. Our definition of hitting time has the advantage that it requires neither the specification of a walker's initial condition nor of an arrival probability threshold. We give full details for the case of quantum walks on the Cayley graphs of Abelian groups. This includes the special cases of quantum walks on the line and on hypercubes.
A quantum walk, \emph{i.e.}, the quantum evolution of a particle on a graph, is termed \emph{scalar} if the internal space of the moving particle (often called the coin) has dimension one. Here, we study the existence of scalar quantum walks on Cayley graphs, which are built from the generators of a group. After deriving a necessary condition on these generators for the existence of a scalar quantum walk, we present a general method to express the evolution operator of the walk, assuming homogeneity of the evolution. We use this necessary condition and the subsequent constructive method to investigate the existence of scalar quantum walks on Cayley graphs of groups presented with two or three generators. In this restricted framework, we classify all groups -- in terms of relations between their generators -- that admit scalar quantum walks, and we also derive the form of the most general evolution operator. Finally, we point out some interesting special cases, and extend our study to a few examples of Cayley graphs built with more than three generators.
Abstract We present an analytical framework to investigate phase dynamics in discrete-time quantum walks, with a focus on geometric phases in both single- and two-particle systems. Leveraging the Cayley-Hamilton theorem, we construct the evolution operator in momentum space and derive exact closed-form solutions for the full time evolution. This allows us to explicitly extract total, dynamic, and most notably, non-adiabatic geometric phases throughout the walk. In the two-particle case, we further examine how initial entanglement influences the accumulation and interplay of these phases. Our results offer a unified and exact approach to understanding the geometric structure underlying quantum walk dynamics, with particular emphasis on non-adiabatic effects.
Abstract. We address the problem of the construction of quantum walks on Cayley graphs. Our main motivation is the relationship between quantum algorithms and quantum walks. In particular, we discuss the choice of the dimension of the local Hilbert space and consider various classes of graphs on which the structure of quantum walks may differ. We completely characterise quantum walks on free groups and present partial results on more general cases. Some examples are given including a family of quantum walks on the hypercube involving a Clifford Algebra. PACS number: 03.67.LxQuantum walks on Cayley graphs 2 1.
We consider the Weyl quantum walk in 3+1 dimensions, that is a discrete-time walk describing a particle with two internal degrees of freedom moving on a Cayley graph of the group [Formula: see text], which in an appropriate regime evolves according to Weyl's equation. The Weyl quantum walk was recently derived as the unique unitary evolution on a Cayley graph of [Formula: see text] that is homogeneous and isotropic. The general solution of the quantum walk evolution is provided here in the position representation, by the analytical expression of the propagator, i.e. transition amplitude from a node of the graph to another node in a finite number of steps. The quantum nature of the walk manifests itself in the interference of the paths on the graph joining the given nodes. The solution is based on the binary encoding of the admissible paths on the graph and on the semigroup structure of the walk transition matrices.This article is part of the themed issue 'Second quantum revolution: foundational questions'.