Perfect state transfer on gcd-graphs over a finite Frobenius ring, I: general theory and results for local rings

TL;DR

We address the PST problem on gcd-graphs over finite Frobenius rings using an algebraic spectral approach that generalizes classical gcd-graph analyses. The core method derives a PST criterion from the $R$-circulant spectrum, expressing eigenvalues via Ramanujan sums tied to a nondegenerate functional $oldsymbol{\psi}$ on $R$. Key contributions include a general PST criterion, necessary conditions restricting PST targets, a complete local-ring PST classification (notably for residue field $ork{ ext{F}}_2$), and a thorough treatment of unitary Cayley graphs, complemented by explicit examples and computational data. The results unify prior arithmetic-GCD graph PST findings and offer a framework for future exploration of PST on broader ring-based graphs.

Abstract

The existence of perfect state transfer (PST) on quantum spin networks is a fundamental problem in mathematics and physics. Various works in the literature have explored PST in graphs with arithmetic origins, such as gcd-graphs over $\mathbb{Z}$ and cubelike graphs. In this article, building on our recent work on gcd-graphs over an arbitrary finite Frobenius ring, we investigate the existence of PST on these graphs. Our approach is algebraic in nature, enabling us to unify various existing results in the literature.

Figures (1)

• Figure 1: Two gcd-graphs over $\mathop{\mathrm{\mathbb{F}}}\nolimits_2[x,y]/(x^2,y^2)$

Theorems & Definitions (41)

• Remark 2.1
• Theorem 2.2
• Definition 3.1
• Corollary 3.2
• proof
• Proposition 3.3
• proof
• Lemma 3.4
• proof
• Theorem 3.5