Encoding Hypergraphs into Quantum States

TL;DR

This paper extends a framework to encode hypergraphs into quantum states, introduces hypergraph states, and explores their properties, relations to other quantum states, and entanglement characteristics.

Contribution

It generalizes the graph-to-quantum-state encoding to hypergraphs, introduces hypergraph states, and analyzes their properties and transformations.

Findings

Hypergraph states are equivalent to real equally weighted states used in quantum algorithms.

Relations among hypergraph states, graph states, and stabilizer states are characterized.

Transformation rules for hypergraph states under local operations are established.

Abstract

Ionicioiu and Spiller [Phys. Rev. A 85, 062313 (2012)] have recently presented an axiomatic framework for mapping graphs to quantum states of a suitable physical system. Based on their study, we first extend the axiomatic framework to hypergraphs by means of modifying its axioms and consistency conditions. Then we use the axiomatic approach to encode hypergraphs into a new family of quantum states, called the hypergraph states. Moreover, we also try to do the followings: (i) to show that real equally weighted states, which occur in Grover and Deutsch-Joza algorithms, are equivalent to hypergraph states; (ii) to describe the relations among hypergraph states, graph states and stabilizer states; (iii) to provide some transformation rules, stated in purely hypergraph theoretical terms, which completely characterize the evolution of hypergraph states under some local operations, including operators in Pauli group and some special local Pauli measurements; and (iv) to investigate some properties of multipartite entanglement of hypergraph states by hypergraph theory.