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A route to quantum computing through the theory of quantum graphs

Farrokh Razavinia

TL;DR

The paper investigates a route to quantum computing through the $C^*$-algebraic theory of quantum graphs, anchored in the coordinate ring $M_q(n)$ and the directed graph $\\mathcal{G}(\\Pi_n)$. It builds Cuntz-Krieger $(\\mathcal{G}(\\Pi_n))$-families to generate graph $C^*$-algebras and a family of quantum channels labeled by Hamiltonian paths, yielding a set of $4n-6$ distinct channels corresponding to a $(4n-6)$-qubit system (with non-entangled states in the presented cases). The approach links noncommutative graph theory to quantum information, offering a route to encode and potentially protect quantum information via noncommutative confusability graphs and exploring connections to quantum error correction. The framework generalizes from the base $2$-qubit example to higher-qubit constructions through a recurrence governing the indexing of CK operators and Hamiltonian-path counts, suggesting a scalable, graph-based encoding scheme.

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 $\mathcal{G}(Π_n)$, and the Cuntz-Krieger $C^*$-graph algebras. Meaningly, we introduce a $(4i-6)$-qubit quantum system by using the Cuntz-Krieger $\mathcal{G}(Π_i)$-families associated to the $4i-6$ distinct Hamiltonian paths of $\mathcal{G}(Π_i)$, 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 $\mathcal{G}(Π_n)$-families.

A route to quantum computing through the theory of quantum graphs

TL;DR

The paper investigates a route to quantum computing through the -algebraic theory of quantum graphs, anchored in the coordinate ring and the directed graph . It builds Cuntz-Krieger -families to generate graph -algebras and a family of quantum channels labeled by Hamiltonian paths, yielding a set of distinct channels corresponding to a -qubit system (with non-entangled states in the presented cases). The approach links noncommutative graph theory to quantum information, offering a route to encode and potentially protect quantum information via noncommutative confusability graphs and exploring connections to quantum error correction. The framework generalizes from the base -qubit example to higher-qubit constructions through a recurrence governing the indexing of CK operators and Hamiltonian-path counts, suggesting a scalable, graph-based encoding scheme.

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 quantum matrix algebra , and the associated directed locally finite graphs , and the Cuntz-Krieger -graph algebras. Meaningly, we introduce a -qubit quantum system by using the Cuntz-Krieger -families associated to the distinct Hamiltonian paths of , for . We also will present a proof of a claim raised in our previous paper concerning the graph -algebra structure and the associated Cuntz-Krieger -families.
Paper Structure (9 sections, 7 theorems, 21 equations, 2 figures)

This paper contains 9 sections, 7 theorems, 21 equations, 2 figures.

Table of Contents

  1. An invitation to quantum graphs
  2. Notation
  3. Quantum Graphs, $C^*$-algebraic approach
  4. $C^*$-Graph Algebras, a rout to quantum ($C^*$-algebraic) graphs
  5. A route to quantum computing through the theory of quantum graphs
  6. 2-qubit quantum system
  7. ($4n-6$)-qubit quantum system
  8. Concluding remarks
  9. Acknowledgement

Key Result

Theorem 3.5

For a quantum channel $\Psi:M_n\to M_m$, and any $\mathcal{X}\in M_n$, with $\Psi(\mathcal{X})=\sum_{i=1}^{r}S_i\mathcal{X}S_{i}^{*}$, the confusability graph of $\Psi$ is the operator system

Figures (2)

  • Figure 1: Directed locally connected graph related to $\Pi_2$
  • Figure 2: Directed locally connected graph $\mathcal{G}(\Pi_3)$

Theorems & Definitions (25)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.5
  • Proposition 4.1
  • Lemma 4.5
  • proof
  • Lemma 4.7
  • proof
  • Remark 4.8
  • ...and 15 more