Control of vertex probability via edge-weight modulation in continuous-time quantum walks

Abstract

Continuous-time quantum walks (CTQWs) provide a versatile framework for exploring quantum transport on graphs. In this work, we investigate how the introduction of edge-weight modulation at a single vertex can suppress its occupation probability. We show that when the edges connected to the root vertex are enhanced by a factor $J$, the probability of detecting the walker at this vertex decays as $1/J^2$, provided the initial state has no components on the vertex itself or its nearest neighbors. We derive the full eigenvalue and eigenvector structure of this system, revealing that the suppression arises from the decoupling of two symmetric line subgraphs and the destructive interference of higher-order contributions. The analysis is extended to tree graphs, where we demonstrate the same scaling behavior and identify the role of local graph geometry in controlling vertex probabilities. These results suggest edge-weight modulation as a mechanism for manipulating transport pathways in CTQWs, with potential applications in quantum information transfer and state engineering, and may serve as a probe of decoherence effects in open quantum systems.

Figures (7)

• Figure 1: Bounded weighted line $L_{2m+1}$: The root vertex $2m+1$ (central) connects to two neighbors with edges weighted by a factor $J$, while all other edges have unit weight. The gray vertices highlight the root vertex and its nearest neighbors.
• Figure 2: Weighted tree graphs: (a) star graph $S_n$, (b) spider graph $S_{n,2}$, and (c) spider graph $S_{n,3}$. Gray vertices highlight the root and its $n$ adjacent vertices, whose connecting edges are weighted by $J$.
• Figure 3: Vertex probabilities $P_v(\tau)$ of CTQWs on spider graphs with root-edge weights $J=1$ (black), $2$ (red), $3$ (blue), and $10$ (green). Top: walks starting from a single leaf vertex $s$. Panels (i)–(iii): (a) $S_{3,2}$ showing probabilities at vertex $s$, the root $r$, and a target leaf $t$. Panels (iv)–(vi): analogous results for (b) $S_{3,3}$. Bottom: walks starting from a balanced superposition of the three leaf vertices $s$. Panels (vii)–(viii) for (c) $S_{3,2}$ and panels (ix)–(x) for (d) $S_{3,3}$ showing probabilities at $s$ and $r$. In both cases, the root probability decreases as $1/J^2$.
• Figure 4: Root probability $P_r(\tau)$ at the root $r=mn+1$ for spider graphs with (a) $m=2$, and (b) $m=3$ for branch numbers $n=3$ (black), $4$ (red), $5$ (blue), and $6$ (green). Solid lines are numerical simulations with $J=10$, and dashed lines are the analytical approximations from Eqs. \ref{['P2n+1approx']} and \ref{['P3n+1approx']}.
• Figure 5: Weighted Cayley trees: (a) $C_{3,2}$ and (b) $C_{3,3}$. Their central-vertex probabilities reproduce those of the spider graphs (a) $S_{3,2}$ and (b) $S_{3,3}$ in panels (ii) and (v) of Fig. \ref{['Fig3']}, respectively. Edges incident to the root carry weight $J$, while all other edges have weight $1/\sqrt{2}$. Gray vertices indicate the root vertex and its nearest neighbors.