Minimising the number of edges in LC-equivalent graph states

TL;DR

This paper introduces methods to find minimal-edge graph states within LC-equivalence classes, reducing resource requirements for quantum state creation, and demonstrates their effectiveness on various examples including repeater states.

Contribution

It presents an integer linear programming approach and a simulated annealing method for edge minimisation, extending to weighted edges and applying to practical quantum communication states.

Findings

Identified new minimum edge representatives for up to 16 qubits.

Proved weighted-edge minimisation is NP-complete.

Applied methods to optimise resource use in quantum repeater states.

Abstract

Graph states are a powerful class of entangled states with numerous applications in quantum communication and quantum computation. Local Clifford (LC) operations that map one graph state to another can alter the structure of the corresponding graphs, including changing the number of edges. Here, we tackle the associated edge-minimisation problem: finding graphs with the minimum number of edges in the LC-equivalence class of a given graph. Such graphs are called minimum edge representatives (MER) and are crucial for minimising the resources required to create a graph state. We leverage Bouchet's algebraic formulation of LC-equivalence to encode the edge-minimisation problem as an integer linear program (EDM-ILP). We further propose a simulated annealing (EDM-SA) approach guided by the local clustering coefficient for edge minimisation. We identify new MERs for graph states with up to 16 qubits by combining EDM-SA and EDM-ILP. We extend the ILP to weighted-edge minimisation, where each edge has an associated weight, and prove that this problem is NP-complete. Finally, we employ our tools to minimise the resources required to create all-photonic generalised repeater graph states using fusion operations.