Related papers: Mixing of Quantum Walks on Generalized Hypercubes
Quantum walks on graphs have shown prioritized benefits and applications in wide areas. In some scenarios, however, it may be more natural and accurate to mandate high-order relationships for hypergraphs, due to the density of information…
A continuous-time quantum walk on a dynamic graph evolves by Schr\"odinger's equation with a sequence of Hamiltonians encoding the edges of the graph. This process is universal for quantum computing, but in general, the dynamic graph that…
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 consider the continuous-time quantum walk defined on the adjacency matrix of a graph. At each instant, the walk defines a mixing matrix which is doubly-stochastic. The average of the mixing matrices contains relevant information about…
A continuous quantum walk on a graph $X$ with adjacency matrix $A$ is specified by the 1-parameter family of unitary matrices $U(t)=\exp(itA)$. These matrices act on the state space of a quantum system, the states of which we may represent…
Quantum and random walks have been shown to be equivalent in the following sense: a time-dependent random walk can be constructed such that its vertex distribution at all time instants is identical to the vertex distribution of any…
A discrete-time quantum walk on a graph is the repeated application of a unitary evolution operator to a Hilbert space corresponding to the graph. If this unitary evolution operator has an associated group of symmetries, then for certain…
A continuous-time quantum walk on a graph is a matrix-valued function $\exp(-\mathtt{i} At)$ over the reals, where $A$ is the adjacency matrix of the graph. Such a quantum walk has universal perfect state transfer if for all vertices $u,v$,…
The expected hitting time of discrete quantum walks on a hypercube (HC) is numerically known to be exponentially shorter than that of their classical analogs in terms of the scaling with the HC dimension. Recent numerical analyses…
Quantum walks on graphs have been shown in certain cases to mix quadratically faster than their classical counterparts. Lifted Markov chains, consisting of a Markov chain on an extended state space which is projected back down to the…
Random walk algorithms are crucial for sampling and approximation problems in statistical physics and theoretical computer science. The mixing property is necessary for Markov chains to approach stationary distributions and is facilitated…
Quantum walks, both discrete and continuous, serve as fundamental tools in quantum information processing with diverse applications. This work introduces a hybrid quantum walk model that integrates the coin mechanism of discrete walks with…
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…
We study perfect state transfer on quantum networks represented by weighted graphs. Our focus is on graphs constructed from the join and related graph operators. Some specific results we prove include: (1) The join of a weighted two-vertex…
The (standard) average mixing matrix of a continuous-time quantum walk is computed by taking the expected value of the mixing matrices of the walk under the uniform sampling distribution on the real line. In this paper we consider…
The theory of rapid mixing random walks plays a fundamental role in the study of modern randomised algorithms. Usually, the mixing time is measured with respect to the worst initial position. It is well known that the presence of…
There are few known exponential speedups for quantum algorithms and these tend to fall into even fewer families. One speedup that has mostly resisted generalization is the use of quantum walks to traverse the welded-tree graph, due to…
We compare discrete-time quantum walks on graphs to their natural classical equivalents, which we argue are lifted Markov chains, that is, classical Markov chains with added memory. We show that these can simulate quantum walks, allowing us…
It is well-known that classical random walks on regular graphs converge to the uniform distribution. Quantum walks, in their various forms, are quantizations of their corresponding classical random walk processes. Gerhardt and Watrous…
Let $X$ be a graph with adjacency matrix $A$. The \textsl{continuous quantum walk} on $X$ is determined by the unitary matrices $U(t)=\exp(itA)$. If $X$ is the complete graph $K_n$ and $a\in V(X)$, then \[1-|U(t)_{a,a}|\le2/n. \] In a…