Type-I and Type-II Fusion Protocols for Weighted Graph States

TL;DR

This paper analyzes two fusion protocols for weighted graph states in measurement-based quantum computation, highlighting their operational differences, success probabilities, and impact on entanglement and state structure.

Contribution

It provides a detailed comparison of Type-I and Type-II fusion operations on weighted graph states, including their success conditions and effects on entanglement.

Findings

Type-I fusion preserves edge weights and success probabilities.

Type-II fusion requires specific weight configurations and has success probability below 50%.

Failure in Type-II fusion destroys the graph structure, affecting MBQC resource quality.

Abstract

Weighted graph states extend standard graph states by associating phases with entangling edges, and may serve as resources for measurement-based quantum computation (MBQC). We analyze how the two main fusion operations, Type-I and Type-II, act on weighted graph states. Type-I fusion operates identically to the unweighted case, merging two one-dimensional weighted graphs, while preserving edge weights and success probabilities. In addition, the pool of 2-qubit weighted graph states can be generated easily by GHZ states or Bell pairs. In contrast, Type-II fusion requires a logical qubit, which can be formed only for specific weight configurations, and with success probability below one-half, which is an obstacle one can avoid. When successful, it fuses the states correctly, but its failure outcomes destroy the structure of the graphs, removing the good-failure feature, known from ordinary graph states. We compute the change in the entanglement entropy of the resulting link due to the fused states being weighted graph states (for generalized fusion), and classify the resulting states of a general non-Bell projection. These results define the practical limits of the fusion-based construction of weighted graph states for MBQC.