Properties of multi-qubit variational quantum states representing weighted graphs and their computing with quantum programming

TL;DR

This paper investigates multi-qubit variational quantum states modeled as weighted graphs, deriving entanglement measures and correlators, and demonstrates their relation to graph structure through theoretical analysis and noisy quantum simulations.

Contribution

It introduces a method to analyze quantum states representing weighted graphs, linking quantum properties to graph structure, and validates findings with noisy quantum simulations.

Findings

Derived geometric measure of entanglement for quantum graph states.

Established relationship between quantum correlators and graph edge weights.

Validated theoretical predictions with noisy quantum simulations on AerSimulator.

Abstract

We study multi-qubit variational quantum states that can be considered as vertex- and edge-weighted graph. These states are constructed as single-layer variational circuits with $RX$ rotations and $RZZ$ entangling gates, corresponding to graphs of arbitrary structure. In general case of quantum graph states of arbitrary structure we derive the geometric measure of entanglement and evaluate quantum correlators. It is shown that these quantities are related to the edge-weight structure around the corresponding vertices in the graph (i.e., edge weights incident to the vertices and vertex weights associated with their closed neighborhoods). In the special case of quantum states representing unweighted graphs, these quantities are related to the degrees of the corresponding vertices in the graph. As an example, we analyze the state associated with the star graph $K_{1,4}$ using noisy quantum computing on the AerSimulator. The results are in good agreement with theoretical predictions. These findings demonstrate a connection between graph structure and quantum properties, enabling the study of properties of classical graphs via quantum computing.