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Laplacian quantum walks on blow-up graphs

Hermie Monterde, Hiranmoy Pal, Steve Kirkland

TL;DR

This work develops a Laplacian-based framework for quantum walks on blow-up graphs, deriving the transition matrix and spectral structure of blow-ups to characterize periodicity, strong cospectrality, LPST, and LPGST. Through Hadamard-diagonalizable graphs and various graph products, joins, paths, and double-star families, it constructs infinite families where every vertex participates in LPST, often even when the base graph does not admit LPST. The paper also analyzes LPGST in blow-ups of paths and double stars, and introduces edge perturbations (matching insertions) as a practical method to induce LPST in certain 4-divisible blow-ups. These results highlight fundamental differences between Laplacian and adjacency quantum walks on blow-ups and provide rich avenues for constructing state-transfer networks. Overall, the work deepens the understanding of Laplacian quantum state transfer on blow-up graphs and offers concrete construction techniques for LPST/LPGST in large, regular graph families.

Abstract

This paper is a sequel to the work of Bhattacharjya et al.\ (J. Phys. A-Math. 57.33: 335303, https://doi.org/10.1088/1751-8121/ad6653) on quantum state transfer on blow-up graphs, where instead of the adjacency matrix, we take the Laplacian matrix as the time-independent Hamiltonian associated with a blow-up graph. We characterize strong cospectrality, periodicity, perfect state transfer (LPST) and pretty good state transfer (LPGST) on blow-up graphs. We present several constructions of blow-up graphs with LPST and produce new infinite families of regular graphs where each vertex is involved in LPST. We also determine LPST and LPGST in blow-ups of classes of trees. Finally, if $n\equiv 0$ (mod 4), then the blow-up of $n$ copies of a graph $G$ has no LPST, but we show that under certain conditions, the addition of an appropriate matching this blow-up graph results in LPST.

Laplacian quantum walks on blow-up graphs

TL;DR

This work develops a Laplacian-based framework for quantum walks on blow-up graphs, deriving the transition matrix and spectral structure of blow-ups to characterize periodicity, strong cospectrality, LPST, and LPGST. Through Hadamard-diagonalizable graphs and various graph products, joins, paths, and double-star families, it constructs infinite families where every vertex participates in LPST, often even when the base graph does not admit LPST. The paper also analyzes LPGST in blow-ups of paths and double stars, and introduces edge perturbations (matching insertions) as a practical method to induce LPST in certain 4-divisible blow-ups. These results highlight fundamental differences between Laplacian and adjacency quantum walks on blow-ups and provide rich avenues for constructing state-transfer networks. Overall, the work deepens the understanding of Laplacian quantum state transfer on blow-up graphs and offers concrete construction techniques for LPST/LPGST in large, regular graph families.

Abstract

This paper is a sequel to the work of Bhattacharjya et al.\ (J. Phys. A-Math. 57.33: 335303, https://doi.org/10.1088/1751-8121/ad6653) on quantum state transfer on blow-up graphs, where instead of the adjacency matrix, we take the Laplacian matrix as the time-independent Hamiltonian associated with a blow-up graph. We characterize strong cospectrality, periodicity, perfect state transfer (LPST) and pretty good state transfer (LPGST) on blow-up graphs. We present several constructions of blow-up graphs with LPST and produce new infinite families of regular graphs where each vertex is involved in LPST. We also determine LPST and LPGST in blow-ups of classes of trees. Finally, if (mod 4), then the blow-up of copies of a graph has no LPST, but we show that under certain conditions, the addition of an appropriate matching this blow-up graph results in LPST.

Paper Structure

This paper contains 11 sections, 26 theorems, 29 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Transition matrix
  3. Periodicity
  4. Strong cospectrality
  5. Pretty good state transfer
  6. Graph products
  7. Joins
  8. Paths
  9. Double stars
  10. Edge perturbation
  11. Future work

Key Result

Proposition 1

Let $u$ and $v$ be two vertices in $G$. The set $T_u\cup T_v$ is a twin set in $\overset{n}{\uplus}~G$ if and only if $u$ and $v$ are false twins in $G.$

Figures (3)

  • Figure 1: The graph $G\vee H$ where $G=O_1\vee (K_2\cup K_2)$ and $H=O_1$ (left); the graph $\overset{2}{\uplus}~(G\vee H)$ with LPST between vertices $(0,u)$ and $(1,u)$ for any vertex $u$ of $G\vee H$ (right)
  • Figure 2: LPST in $\overset{4}{\uplus}~K_3$ between the end vertices of the dashed edges.
  • Figure 3: LPST in $\overset{4}{\uplus}~C_4$ between the end vertices of the dashed edges.

Theorems & Definitions (42)

  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Theorem 2
  • proof
  • Corollary 3
  • proof
  • Theorem 3
  • ...and 32 more