Non-Abelian Topological Approach to Non-Locality of a Hypergraph State

Vesna Berec

doi:10.3390/e17053376

Abstract

We present a theoretical study of new families of stochastic complex information modules encoded in the hypergraph states which are defined by the fractional entropic descriptor. The essential connection between the Lyapunov exponents and d-regular hypergraph fractal set is elucidated. To further resolve the divergence in the complexity of classical and quantum representation of a hypergraph, we have investigated the notion of non-amenability and its relation to combinatorics of dynamical self-organization for the case of fractal system of free group on finite generators. The exact relation between notion of hypergraph non-locality and quantum encoding through system sets of specified non-Abelian fractal geometric structures is presented. Obtained results give important impetus towards designing of approximation algorithms for chip imprinted circuits in scalable quantum information systems.

Highlights

The field of algebraic topology has passed an essential evolution in attaching algebraic objects to topological spaces starting from the simple graph models associated to vertices and edges in terms of Laplacian matrices, leading to higher-order dimensional structures modified through simplices or cliques, finalizing with an elegant mature representation of non-local hypergraph states [1]
We present a theoretical study of new families of stochastic complex information modules encoded in the hypergraph states which are defined by the fractional entropic descriptor
As we introduce some of the basic notations related to the hypergraph based fractal set, without loss of generality we assume topology embedded in 3, where a hypergraph [25] represents a compact connected Hausdorff space represented by a subset G (V, E ) of vertex elements V = {v1, v2,..., vn } and edges E ⊂ V, E ≠ 0

Summary

Introduction

The field of algebraic topology has passed an essential evolution in attaching algebraic objects to topological spaces starting from the simple graph models associated to vertices and edges in terms of Laplacian matrices, leading to higher-order dimensional structures modified through simplices or cliques, finalizing with an elegant mature representation of non-local hypergraph states [1]. In general sense the topological entropy represents quantitative measure of complexity for continuous map defined on compact metric spaces of dynamical system [2]. In this study we introduce certain situations where the non-local correlations arise from the nonAbelian structure of topological algebraic set systems, addressing at the same time the divergence in dimension and complexity measures of classical and quantum hypergraph representation.

Non-Abelian Statistics over Hypergraph Fractal Set

Topological Entropy and Periodic Orbit Growth for Hypergraph Fractal Set

Rényi Topological Entropy and Lyapunov Exponents for Non-Abelian Fractal Set

Underlying Non-Locality of F2 —Hypergraph State

Quantum Hypergraph State

Conclusions