Encoding Hypergraphs into Quantum States

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.

16 pages, 2 figures

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Cited by in corpus (43)

• Quantum Hypergraph States
• Error protected qubits in a silicon photonic chip
• Efficient Verification of Hypergraph States
• General framework for verifying pure quantum states in the adversarial scenario
• Statistical Methods for Quantum State Verification and Fidelity Estimation
• Extreme violation of local realism in quantum hypergraph states
• Changing the circuit-depth complexity of measurement-based quantum computation with hypergraph states
• Quantum Experiments and Hypergraphs: Multi-Photon Sources for Quantum Interference, Quantum Computation and Quantum Entanglement
• Multipartite entanglement detection for hypergraph states
• Local unitary symmetries of hypergraph states
• Efficient verification of quantum gates with local operations
• Magic of quantum hypergraph states
• Graph states and local unitary transformations beyond local Clifford operations
• Hypergraph Simplification: Linking the Path-sum Approach to the ZH-calculus
• Qudit Hypergraph States
• Entanglement of random hypergraph states
• Dual correspondence between classical spin models and quantum CSS states
• Relationship among locally maximally entanglable states, W states and hypergraph states under local unitary transformations
• Locally inequivalent four qubit hypergraph states
• Graphical description of unitary transformations on hypergraph states
• Qudit hypergraph states and their properties
• Multipartite Entanglement and Hypergraph states of three qubits
• Analysis of quantum error correction with symmetric hypergraph states
• A strong-weak coupling duality between two perturbed quantum many-body systems: CSS codes and Ising-like systems
• Calculating Concentratable Entanglement in Graph States
• Symmetric hypergraph states: Entanglement quantification and robust Bell nonlocality
• Quantum Hypergraph States in Continuous Variables
• Local Pauli stabilizers of symmetric hypergraph states
• Universal quantum gates, artificial neurons and pattern recognition simulated by \textit{LC} resonators
• Classical criticality establishes quantum topological order
• Entanglement Purification of Hypergraph States
• Quantum computation by teleportation and symmetry
• Multiple multi-control unitary operations: implementation and applications
• Multipartite entanglement sudden death and birth in randomized hypergraph states
• Group structures and representations of graph states
• Entropic measure and hypergraph states
• Multipartite States under Elementary Local Operations
• Graphical Framework for Non-Gaussian Quantum States
• Finite-Function-Encoding Quantum States
• Randomized hypergraph states and their entanglement properties
• Entanglement of $\pi$-LME states and the SAT problem
• Phase space dynamics and control of the quantum particles associated to hypergraph states
• Non-symmetric GHZ states: weighted hypergraph and controlled-unitary graph representations