Generalized Graph States Based on Hadamard Matrices

TL;DR

This paper introduces a new class of generalized graph states based on Hadamard matrices, expanding the framework of quantum states used in quantum information processing and exploring their entanglement and symmetry properties.

Contribution

It proposes a novel generalization of graph states using Hadamard matrices and studies their entanglement structure, local symmetries, and potential for constructing quantum codes beyond stabilizer codes.

Findings

All generalized graph states are maximally mixed locally.

The equivalence of Hadamard matrices relates to local equivalence of states.

Framework extends to non-stabilizer quantum codes.

Abstract

Graph states are widely used in quantum information theory, including entanglement theory, quantum error correction, and one-way quantum computing. Graph states have a nice structure related to a certain graph, which is given by either a stabilizer group or an encoding circuit, both can be directly given by the graph. To generalize graph states, whose stabilizer groups are abelian subgroups of the Pauli group, one approach taken is to study non-abelian stabilizers. In this work, we propose to generalize graph states based on the encoding circuit, which is completely determined by the graph and a Hadamard matrix. We study the entanglement structures of these generalized graph states, and show that they are all maximally mixed locally. We also explore the relationship between the equivalence of Hadamard matrices and local equivalence of the corresponding generalized graph states. This leads to a natural generalization of the Pauli $(X,Z)$ pairs, which characterizes the local symmetries of these generalized graph states. Our approach is also naturally generalized to construct graph quantum codes which are beyond stabilizer codes.