bibkey: coutinhobaptista2023normalized
authors: Gabriel Coutinho; Pedro Ferreira Baptista
year: 2023
title: "Quantum walks in the normalized Laplacian"
doi: 10.1080/03081087.2023.2236282
url: https://arxiv.org/abs/2209.04668v1
claim: "The paper characterizes perfect state transfer in the continuous quantum walk generated by the normalized Laplacian, restricts the eigenvalue supports of transferring vertices in trees, and conjectures that no tree other than the paths on two and three vertices admits perfect state transfer in this model."
strata_touched:
  - D5/S3/Quantum/Dynamics/NormalizedLaplacianTreeStateTransfer
license: citation-only
triage: anchor

Quantum walks in the normalized Laplacian

G. Coutinho and P. F. Baptista, arXiv:2209.04668v1 (2022); Linear Multilinear Algebra 72 (2024) 2123–2133, published online 2023. Quotations are from the v1 TeX source.

The normalized Laplacian and the walk:

Let $G$ be a connected graph on two or more vertices. Let $A$ be its adjacency matrix and $D$ the diagonal matrix with the degrees of its vertices in its entries. The combinatorial Laplacian matrix is defined as $L=D-A$, and the normalized Laplacian as $\mathcal{L}=D^{-1/2}L{D}^{-1/2}.$

$U(t)=\exp (it\mathcal{L})={\sum }_{r=0}^d{e}^{it{\theta }_r}{F}_r,$

Perfect state transfer:

Recall that the quantum walk transition matrix is given by $U(t)=\exp (it\mathcal{L})$. We say that perfect state transfer happens between vertices $a$ and $b$ at time $\tau$ if there is a complex unit $\lambda$ so that $U(\tau )|a⟩=\lambda |b⟩.$

The conjecture:

For trees, we were not able to show a complete analogous result to the combinatorial Laplacian case — that is, that perfect state transfer does not occur. We were however able to significantly restrict the possible cases, and therefore we conjecture that apart from the paths on $2$ and $3$ vertices, no tree admits perfect state transfer according to the normalized Laplacian.

Verified locator

Source: https://arxiv.org/abs/2209.04668v1

• URL: https://arxiv.org/abs/2209.04668v1 (source norm-laplacian-v2.tex, md5 b274bca76e7a9e881ea5b092556075a9):
  • the normalized Laplacian and the walk, l. 90–96;
  • perfect state transfer, l. 161–164;
  • the conjecture, l. 427.
• DOI: 10.1080/03081087.2023.2236282 (https://doi.org/10.1080/03081087.2023.2236282).