Group structures and representations of graph states

TL;DR

This paper introduces X-chains in graph states, revealing their group structure and applications in state representation, measurement outcome analysis, error correction, and efficient overlap computation.

Contribution

It defines X-chains for graph states, develops methods for their efficient determination, and demonstrates their utility in state analysis and quantum information tasks.

Findings

X-chains form a group structure enabling explicit state representation.

Different X-chain groups lead to distinguishable measurement outcome distributions.

Overlap of graph states can be computed efficiently using X-chains.

Abstract

A special configuration of graph state stabilizers, which contains only Pauli $\sigma_X$ operators, is studied. The vertex sets $\xi$ associated with such configurations are defined as what we call X-chains of graph states. The X-chains of a general graph state can be determined efficiently. They form a group structure such that one can obtain the explicit representation of graph states in the X-basis via the so-called X-chain factorization diagram. We show that graph states with different X-chain groups can have different probability distributions of X-measurement outcomes, which allows one to distinguish certain graph states with X-measurements. We provide an approach to find the Schmidt decomposition of graph states in the X-basis. The existence of X-chains in a subsystem facilitates error correction in the entanglement localization of graph states. In all of these applications, the difficulty of the task decreases with increasing number of X-chains. Furthermore, we show that the overlap of two graph states can be efficiently determined via X-chains, while its computational complexity with other known methods increases exponentially.