In quantum computing, a graph state is a special type of multi-qubit state that can be represented by a graph.
Each qubit is represented by a vertex of the graph, and there is an edge between every interacting pair of qubits.
In particular, they are a convenient way of representing certain types of entangled states.
Graph states are useful in quantum error-correcting codes, entanglement measurement and purification and for characterization of computational resources in measurement based quantum computing models.
A graph state is a particular case of a stabilizer state as well as a 2-uniform hypergraph state, a generalization where the edges have cardinality between 1 and N. Quantum graph states can be defined in two equivalent ways: through the notion of quantum circuits and stabilizer formalism.
, the corresponding graph state is defined as where
is the controlled-Z interaction between the two vertices (corresponding to two qubits)
An alternative and equivalent definition is the following, which makes use of the stabilizer formalism.
A proof of the equivalence of the two definitions can be found in.
[1][2] The corresponding quantum state is The corresponding quantum state is Observe that
are locally equivalent to each other, i.e., can be mapped to each other by applying one-qubit unitary transformations.
on the middle qubit, maps the stabilizer group of one into that of the other.
Two graph states are called locally equivalent if one can be converted into the other by local unitary gates.
If the conversion from one state to the other can be performed by local gates from the Clifford group, the two states are called locally Clifford equivalent.
If and only if two graph states are locally Clifford equivalent, one graph can be converted into the other by a sequence of so-called "local complementations".
[3] This gives a useful tool for studying local Clifford equivalence by a simple graph-manipulation rule and corresponding equivalence classes of graph states have been studied in Refs.
[1][4][5] However, local Clifford equivalence of graph states only coincides with local unitary equivalence for small graph states[1] and is generally not identical.
[6] After a graph state was created in an experiment, it is important to verify that indeed, an entangled quantum state has been created.
for a nontrivial graph state corresponding to a connected graph, then the state
[7] [8] Thus, one can obtain an entanglement witness detecting entanglement close the graph states as
signals genuine multiparticle entanglement.
It has to be decomposed to a sum of correlations terms, which can then be measured.
However, for large systems this approach can be difficult.
There are also entanglement witnesses that work in very large systems, and they also detect genuine multipartite entanglement close to graph states.
Here, the graph state itself has to be genuine multipartite entangled, that is, it has to correspond to a connected graph.
The witnesses need only the minimal two local measurement settings for graph states corresponding to two-colorable graphs.
[7] [8] Similar conditions can also be used to put a lower bound on the fidelity with respect to an ideal graph state.
[8] These criteria have been used first in an experiment realizing four-qubit cluster states with photons.
[9] These approaches have also been used to propose methods for detecting entanglement in a smaller part of a large cluster state or graph state realized in optical lattices.
[11] [12] [13] All these entanglement conditions and Bell inequalities are based on the stabilizer formalism.