Pulsation of quantum walk on finite graph
Abstract
We study discrete-time quantum walks on weighted graphs, where every edge in an edge cut set is assigned a small weight ε>0. The parameter ε represents the strength of the connections through the cut edges: as ε→ 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 ε and on the time scale t=Θ(ε−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.
Related
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- Walk/Zeta correspondence for quantum walks on path graphs: a pre-geometric structure implied by a generalized reflection principle
- Evolution of weights on a connected finite graph
BibTeX
@misc{hosaka2026pulsation,
title = {{Pulsation of quantum walk on finite graph}},
author = {Taisuke Hosaka and Etsuo Segawa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07768},
eprint = {2610.07768},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07768},
}
RIS
TY - UNPB
TI - Pulsation of quantum walk on finite graph
AU - Taisuke Hosaka
AU - Etsuo Segawa
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.07768
UR - https://arxiv.org/abs/2610.07768
AB - We study discrete-time quantum walks on weighted graphs, where every edge in an edge cut set is assigned a small weight ε>0. The parameter ε represents the strength of the connections through the cut edges: as ε→ 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 ε and on the time scale t=Θ(ε^−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.
ER -