Transforming graph states via Bell state measurements

TL;DR

This paper develops comprehensive rules for transforming graph states using Bell state measurements, enabling advanced quantum computing applications with photonic systems.

Contribution

It provides a complete set of graph transformation rules for all types of rotated type-II fusions, including visualization and practical application examples.

Findings

Derived all graph transformation rules for rotated type-II fusion

Visualized transformations using Venn diagrams of local neighborhoods

Demonstrated applications in constructing graph codes and simulating fusion networks

Abstract

Graph states are key resources for measurement-based quantum computing, which is particularly promising for photonic systems. Fusions are probabilistic Bell state measurements, measuring pairs of parity operators of two qubits. Fusions can be used to connect/entangle different graph states, making them a powerful resource for measurement-based and related fusion-based quantum computing. There are several different graph structures and types of Bell state measurements, yet the associated graph transformations have only been analyzed for specific cases. Here, we provide a full set of graph transformation rules and give an intuitive visualization based on Venn diagrams of local neighborhoods of graph nodes. We derive these graph transformations for all types of rotated type-II fusion, showing that there are five different fusion success cases. Finally, we give application examples of the derived graph transformation rules and show that they can be used to construct graph codes or simulate fusion networks.