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Spectral analysis of hierarchical continuous-time quantum walks

Jirô Akahori, Yusuke Ide, Tomoki Kato, Norio Konno, Shuhei Mano, Akihiro Narimatsu

TL;DR

This paper develops a hierarchical framework for multi-walker quantum walks by combining a global and local walker to form hierarchical discrete-time random walks ($hDTRW$) and the corresponding hierarchical continuous-time quantum walks ($hCTQW$). It then defines multi-dimensional CTQW ($mCTQW$) as the marginal distributions of local walkers under the $hCTQW$, and provides a detailed spectral analysis with a block-structured decomposition of the transition operator $P_G$ and the time-evolution operator $U_G(t)$, including explicit results for the complete-graph-with-self-loops case $H=\overline{K_{d+1}}$. The work shows how marginal distributions can factor or reduce to product forms under certain initial-state overlap conditions, connecting to IdeKonnoNarimatsu2025 and offering a scalable approach to analyze complex quantum-walk systems. Overall, the framework advances understanding of spectral structures in hierarchical and multi-dimensional quantum walks with potential applications in quantum information and network science.

Abstract

In this paper, we introduce hierarchical random walks at first. In this model, we use two types of random walkers, {global and local} walkers. The global walker chooses a local walker at every step, then the chosen local walker moves a single step. After that we construct the corresponding continuous-time quantum walks and discuss its spectral structures. Then we define multi-dimensional continuous-time quantum walk by taking a marginal distribution respect to the global walker.

Spectral analysis of hierarchical continuous-time quantum walks

TL;DR

This paper develops a hierarchical framework for multi-walker quantum walks by combining a global and local walker to form hierarchical discrete-time random walks () and the corresponding hierarchical continuous-time quantum walks (). It then defines multi-dimensional CTQW () as the marginal distributions of local walkers under the , and provides a detailed spectral analysis with a block-structured decomposition of the transition operator and the time-evolution operator , including explicit results for the complete-graph-with-self-loops case . The work shows how marginal distributions can factor or reduce to product forms under certain initial-state overlap conditions, connecting to IdeKonnoNarimatsu2025 and offering a scalable approach to analyze complex quantum-walk systems. Overall, the framework advances understanding of spectral structures in hierarchical and multi-dimensional quantum walks with potential applications in quantum information and network science.

Abstract

In this paper, we introduce hierarchical random walks at first. In this model, we use two types of random walkers, {global and local} walkers. The global walker chooses a local walker at every step, then the chosen local walker moves a single step. After that we construct the corresponding continuous-time quantum walks and discuss its spectral structures. Then we define multi-dimensional continuous-time quantum walk by taking a marginal distribution respect to the global walker.
Paper Structure (3 sections, 4 theorems, 36 equations)

This paper contains 3 sections, 4 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Hierarchical continuous-time quantum walk
  3. Multi-dimensional continuous-time quantum walk

Key Result

Proposition 2.1

If we obtain the eigenvalue $\lambda _{\ell }^{(\ell^{(0)}, \ldots , \ell^{(d)})}$ and the corresponding eigenvector $|v_{\ell }^{(\ell^{(0)}, \ldots , \ell^{(d)})}\rangle$ the matrix $P_{H} \Lambda ^{(\ell^{(0)}, \ldots , \ell^{(d)})}$ i.e. for $\ell=0,\ldots ,d$, then we have the eigenvalue and the corresponding eigenvectors of $P_{G}$ as

Theorems & Definitions (4)

  • Proposition 2.2
  • Theorem 2.3
  • Lemma 3.1
  • Theorem 3.2