Relation of curvature and torsion of weighted graph states with graph properties and its studies on a quantum computer

TL;DR

This paper explores how the geometric properties of weighted graph states, such as curvature and torsion, relate to graph characteristics and demonstrates their analysis using quantum computing on IBM's platform.

Contribution

It introduces a novel connection between graph geometric properties and weighted degrees, and applies quantum programming to analyze these properties on a quantum computer.

Findings

Velocity of quantum evolution linked to weighted degrees of nodes.

Curvature depends on sums of weighted degrees raised to second and fourth powers.

Torsion relates to products of weights in triangular subgraphs.

Abstract

Quantum states of spin systems that can be represented with weighted graphs $G(V, E)$ are studied. The geometrical characteristics of these states are examined. We find that the velocity of quantum evolution is determined by the sum of the weighted degrees of the nodes in the graph, constructed by raising to the second power the weights in $G(V, E)$. The curvature depends on the sum of the weighted degrees of nodes in graphs constructed by raising the weights in $G(V, E)$ to the second and fourth powers. It also depends on the sum of the products of the weights of edges forming squares in graph $G(V, E)$. The torsion in addition is related to the sum of the products of the weights of edges in graph $G(V, E)$ forming triangles $S_3$. Geometric properties of quantum graph states and the sum of the weighted degrees of nodes have been calculated with quantum programming on IBM's quantum computer for the case of a spin chain.