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Quantum hypergraph states: a review

Vinícius Salem

TL;DR

The definition of hypergraph states is reviewed and their main applications so far are reviewed, both in discrete-variable and continuous-variable quantum information.

Abstract

Quantum hypergraph states emerged in the literature as a generalization of graph states, and since then, considerable progress has been made toward implementing this class of genuine multipartite entangled states for quantum information and computation. Here, we review the definition of hypergraph states and their main applications so far, both in discrete-variable and continuous-variable quantum information.

Quantum hypergraph states: a review

TL;DR

The definition of hypergraph states is reviewed and their main applications so far are reviewed, both in discrete-variable and continuous-variable quantum information.

Abstract

Quantum hypergraph states emerged in the literature as a generalization of graph states, and since then, considerable progress has been made toward implementing this class of genuine multipartite entangled states for quantum information and computation. Here, we review the definition of hypergraph states and their main applications so far, both in discrete-variable and continuous-variable quantum information.

Paper Structure

This paper contains 22 sections, 98 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Quantum Entanglement
  3. Bipartite Entanglement
  4. Multiparticle entanglement
  5. Definitions
  6. Basic hypergraph theory
  7. Quantum hypergraph states
  8. Qudit hypergraph states
  9. Mixed Hypergraph States
  10. LME States
  11. The stabilizer formalism
  12. Magic or Nonstabilizer states
  13. The generalized stabilizer formalism
  14. Local unitary operations and equivalence classes of entanglement
  15. Equivalence classes under local operations
  16. ...and 7 more sections

Figures (11)

  • Figure 1: Hypergraphs are generalizations of graphs. (a) A regular graph with 5 vertices, (b) A single-hyperedge hypergraph, (c) the same trivial hypergraph combined with a star graph, (d) a 4-uniform hypergraph representing a GHZ state.
  • Figure 2: Illustration of the correspondence between a non-uniform hypergraph state $H_5$, on the left, and its associated quantum state, represented in the five-level quantum circuit $C_5$.
  • Figure 3: $3$-uniform (a-d) Clover hypergraphs $Cl_n$ and (e-g) the Hyperflower hypergraphs $Fl_n$.
  • Figure 4: Examples of multi-hypergraph states equivalent to qudit hypergraph states: a) a qudit graph state, b) qudit graph state, c) a non-uniform multi-hypergraph.
  • Figure 5: Randomization procedure for the three-qubit hypergraph state and its respective subhypergraphs.
  • ...and 6 more figures

Theorems & Definitions (15)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • ...and 5 more