Fig. 1: Enumeration of all four-qubit hypergraph states under local unitary transformation.
From: Demonstration of hypergraph-state quantum information processing
a Topology of a four-qubit graph state, and b a four-qubit hypergraph state. Vertices (V) represent qubits (Q1--Q4), and edges (E, black lines) represent entangling interactions between two qubits, and hyperedges (HE, colored close shapes) represent entangling interactions between multiple qubits. Right plots: Quantum circuits for the preparations of graph state and hypergraph state, shown in the left plots respectively. Vertical colored lines connecting red dots represent the CZ, CCZ and CCCZ entangling gates. c An example to show the principle of local unitary (LU) equivalence. The hypergraphs are locally equivalent if they can be mutually transformed by repeatedly applying the local unitary operations on single-qubits, e.g, local Pauli operations X(k) and Z(k) on the qubit k. The X(k) operation on the qubit k removes or adds these hyperedges in E(k) depending on whether they exist already or not, where E(k) represents all hyperedges that contain qubit k but removing qubit k out. The Z(k) operation on the qubit k removes the one-edge on the qubit k. step1 to step 2: the X3 operation on Q3; step2 to step 3: the X1 operation on Q1; step3 to step4: the one-edge is removed by a Z2 operation. d Enumeration of all 27 four-qubit hypergraph states that are equivalent under LU transformation. The first two refer to two classes of star (\(\left\vert S\right\rangle\)) and line (\(\left\vert L\right\rangle\)) graph states.
