A route to quantum computing through the theory of quantum graphs

TL;DR

This paper explores the connection between quantum graphs and quantum computing by introducing new mathematical structures based on quantum matrix algebras, graph $C^*$-algebras, and Hamiltonian paths to model quantum systems.

Contribution

It introduces a novel framework linking quantum graphs, $C^*$-algebras, and quantum systems, expanding the mathematical tools for quantum computing research.

Findings

Defined a $(4i-6)$-qubit quantum system using Cuntz-Krieger graph families.

Established a proof relating graph $C^*$-algebra structures to quantum systems.

Connected quantum graph theory with quantum computing principles.

Abstract

Based on our previous works, and in order to relate them with the theory of quantum graphs and the quantum computing principles, we once again try to introduce some newly developed technical structures just by relying on our toy example, the coordinate ring of $n\times n$ quantum matrix algebra $M_q(n)$, and the associated directed locally finite graphs $\mathcal{G}(\Pi_n)$, and the Cuntz-Krieger $C^*$-graph algebras. Meaningly, we introduce a $(4i-6)$-qubit quantum system by using the Cuntz-Krieger $\mathcal{G}(\Pi_i)$-families associated to the $4i-6$ distinct Hamiltonian paths of $\mathcal{G}(\Pi_i)$, for $i\in\{2,\cdots,n\}$. We also will present a proof of a claim raised in our previous paper concerning the graph $C^*$-algebra structure and the associated Cuntz-Krieger $\mathcal{G}(\Pi_n)$-families.