Limiting distributions of ergodic continuous-time quantum walks on periodic graphs

TL;DR

This work analyzes ergodicity and limiting distributions for continuous-time quantum walks on $\mathbb{Z}^d$-periodic graphs (crystals) using Bloch-Floquet theory. A Floquet-based ergodicity criterion yields a limiting density $d(p,q)=\int_{\mathbb{T}^d}\sum_{s=1}^{\nu'}|P_{E_s}(\theta)(p,q)|^2\,d\theta$ when the nonresonance condition holds, and the authors compute explicit weights for various fiber graphs in Cartesian products, including cycles, paths, stars, and hypercubes. They show that ergodicity can hold in the horizontal direction and, in some cases, also in sectional directions, but in other families (e.g., hypercubes, stars) mass concentrates near the starting layer, highlighting a marked quantum departure from the uniform limiting behavior of classical random walks on regular graphs. The results provide exact, graph-dependent limiting distributions useful for understanding quantum transport and spectral-geometry on periodic structures. The contrast between uniform classical limits and nonuniform quantum limits illuminates delocalization patterns across graph families and product constructions.

Abstract

In this expository note, we study several families of periodic graphs which satisfy a sufficient condition for the ergodicity of the associated continuous-time quantum walk. For these graphs, we compute the limiting distribution of the walk explicitly. We uncover interesting behavior where in some families, the walk is ergodic in both horizontal and sectional directions, while in others, ergodicity only holds in the horizontal (large N) direction. We compare this to the limiting distribution of classical random walks on the same graphs.

Figures (3)

• Figure 1: The limiting distribution of the continuous-time quantum walk on graphs of the form $\mathbb{Z} \mathop\square G_F$ for $G_F$ as the cycle $C_6$ (left) and the path $P_6$ (right). In both figures, the walk starts at a red vertex, and it is the vertex where most of the weight concentrates. All the green vertices have the same weight, lower than the red vertices.
• Figure 2: The graph $\mathbb{Z}\mathop\square K_{3,1}$. No matter how many edges the star $K_{\nu,1}$ has, if we launch the walk from a red vertex, half the mass spreads over the line of red vertices, while the remainder gets equidistributed over the leaves. The situation is more dramatic if the walk is launched from a leaf vertex: if $\nu=10$, then $0.815$ of the mass spreads over the layer of this leaf.
• Figure 3: Diagram showing the weight distribution of the continuous-time quantum walk on graphs of the form $\Gamma_0 \mathop\square G_F$ for $G_F$ as Petersen graph (left) and the complete bipartite graph $K_{4,4}$(right). In both the figures, the walk starts at the red vertex, and it is the vertex where most of the weight concentrates. The purple vertices have the second-highest weight and the green vertices, the lowest.

Theorems & Definitions (12)

• Theorem 1.1: From BdMS
• Proposition 1.2: Case of Cartesian and Tensor Products
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Corollary 1.7
• proof : Proof of Proposition \ref{['prp:mcks']}
• proof : Proof of Theorem \ref{['thm:cyc']}
• proof : Proof of Theorem \ref{['thm:path']}