We study discrete-time quantum walks on weighted graphs, where every edge in an edge cut set is assigned a small weight $\epsilon>0$. The parameter $\epsilon$ represents the strength of the connections through the cut edges: as $\epsilon \to 0$, these connections vanish, and the graph decomposes into the connected components obtained by removing the cut edges. We refer to these weighted cut edges as {\it weak edges}. We show that, for sufficiently small $\epsilon$ and on the time scale $t=\Theta(\epsilon^{-1/2})$, the finding probabilities between the connected components are asymptotically described by a continuous-time wave equation on a reduced graph, whose vertices represent the connected components and whose edges represent the weak edges connecting them. The wave equation is governed by a symmetric weighted Laplacian determined by the reduced graph and the number of arcs contained in the components. Consequently, finding probabilities of leading-term are independent of the detailed internal structures of the components. We further show that this wave equation has the same form as Newton's equation of motion for a classical spring-mass system on the reduced graph.
We study a generalization of undirected $st$-connectivity to graphs whose edges carry quantum operations. Let $G=(V,E)$ be an undirected graph on $n$ vertices in which each edge $\{u,v\}$ is labeled by a unitary $U_{uv}\in\mathbb{C}^{k\times k}$, with $U_{vu}=U_{uv}^\dagger$. We assume the labels form a \emph{flat} con...
Stacey Jeffery, T. Osborne, Galina Pass· 1 citation
The trace of the average mixing matrix of a quantum walk measures the"laziness"of the walk: the higher the trace, the more likely that the walker returns home in the long run. In this paper, we develop tools to study this graph invariant arising from Laplacian quantum walks. It is known that the complete graph $K_n$ is...
Amulya Mohan, Christino Tamon, Yi-Chi Xu et al.· 0 citations
For a Dirac operator $D$ on a finite weighted graph, let $d^D$ denote the associated noncommutative (Connes) distance. We show that deleting an edge can actually decrease the noncommutative distance between two vertices in the graph. The smallest example of this phenomenon comes from a weighted 4-cycle, and the case of...
Let $U_X(t)$ be the transition matrix of a quantum walk on a graph $X$ relative to its adjacency matrix $A$ or the Laplacian matrix $L$. This paper investigates the behavior of quantum walks under Cartesian products, joins, and graph complements. We have two main goals. First, we characterize the conditions such that p...
Joy Cooper, Homer De Vera, Hermie Monterde et al.· 0 citations
Consider an infinite edge-weighted graph satisfying an isoperimetric inequality of the type $\|\partial A\|\geq C|A|^\alpha$ for some $\alpha,C>0$, where $\|\partial A\|$ denotes the weighted size of the edge boundary of $A$. We prove that, for $C$ large enough depending on $\alpha$, if each edge is open independently...
Ivailo Hartarsky, Franco Severo, Augusto Teixeira· 1 citation
The edge complexity of a weighted graph is the smallest ratio of the Fourier $\ell^1$ and $\ell^2$ norms of its adjacency matrix over all vertex labelings. We study the difficulty of finding this minimum by relating it to a graph symmetry. Adding a universal vertex with sufficiently large incident weight produces an ex...
Vishal Gupta, A. Iosevich· 0 citations
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