Related papers: Cellular Algebras and Graph Invariants Based on Qu…
In this paper, we introduce a quantum walk whose local scattering at each vertex is denoted by a unitary circulant matrix; namely the circulant quantum walk. We also introduce another quantum walk induced by the circulant quantum walk;…
Quantum walks are promising tools based on classical random walks, with plenty of applications such as many variants of optimization. Here we introduce the semiclassical walks in discrete time, which are algorithms that combines classical…
Quantum walks provide a natural framework to approach graph problems with quantum computers, exhibiting speedups over their classical counterparts for tasks such as the search for marked nodes or the prediction of missing links.…
We introduce a continuous-time quantum walk on an ultrametric space corresponding to the set of p-adic integers and compute its time-averaged probability distribution. It is shown that localization occurs for any location of the ultrametric…
It is known that graphs cellularly embedded into surfaces are equivalent to ribbon graphs. In this work, we generalize this statement to broader classes of graphs and surfaces. Half-edge graphs extend abstract graphs and are useful in…
For a quantum walk on a graph, there exist many kinds of operators for the discrete-time evolution. We give a general relation between the characteristic polynomial of the evolution matrix of a quantum walk on edges and that of a kind of…
In this note, we discuss a general definition of quantum random walks on graphs and illustrate with a simple graph the possibility of very different behavior between a classical random walk and its quantum analogue. In this graph,…
Universal quantum computation can be realised using both continuous-time and discrete-time quantum walks. We present a version based on single particle discrete-time quantum walk to realize multi-qubit computation tasks. The scalability of…
We investigate a connection between a property of the distribution and a conserved quantity for the reversible cellular automaton derived from a discrete-time quantum walk in one dimension. As a corollary, we give a detailed information of…
Quantum walks have frequently envisioned the behavior of a quantum state traversing a classically defined, generally finite, graph structure. While this approach has already generated significant results, it imposes a strong assumption: all…
Using spectral graph theory, we show how to obtain inequalities for the number of walks in graphs from nonnegative polynomials and present a new family of such inequalities.
We examine the time dependent amplitude $ \phi_{j}\left( t\right)$ at each vertex $j$ of a continuous-time quantum walk on the cycle $C_{n}$. In many cases the Lissajous curve of the real vs. imaginary parts of each $ \phi_{j}\left(…
In this paper we isolate the combinatorial property responsible (at least in part) for the computational speedups recently observed in some quantum walk algorithms. We find that continuous-time quantum walks can exploit the covering space…
We report on results about a study of algebraic graph invariants, based on computer exploration, and motivated by graph-isomorphism and reconstruction problems.
We define the fundamental quandle of a spatial graph and several invariants derived from it. In the category of graph tangles, we define an invariant based on the walks in the graph and cocycles from nonabelian quandle cohomology.
We study continuous-time quantum walks on normal Cayley graphs of certain non-abelian groups, called extraspecial groups. By applying general results for graphs in association schemes we determine the precise conditions for perfect state…
Quantum walks (QWs) describe particles evolving coherently on a lattice. The internal degree of freedom corresponds to a Hilbert space, called coin system. We consider QWs on Cayley graphs of some group $G$. In the literature,…
The $k$-th power of the adjacency matrix of a simple undirected graph represents the number of walks with length $k$ between pairs of nodes. As a walk where no node repeats, a path is a walk where each node is only visited once. The set of…
We build interacting Fock spaces from association schemes and set up quantum walks on the resulting regular graphs (distance-regular and distance-transitive). The construction is valid for growing graphs and the interacting Fock space is…
Starting from the working hypothesis that both physics and the corresponding mathematics and in particular geometry have to be described by means of discrete concepts on the Planck-scale, one of the many problems one has to face in this…