Rahul Sharma (Editor)

Graph state

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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.

Contents

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.

Formal definition

Given a graph G = (V, E), with the set of vertices V and the set of edges E, the corresponding graph state is defined as

| G ⟩ = ∏ ( a , b ) ∈ E U { a , b } | + ⟩ ⊗ V

where the operator U { a , b } is the controlled-Z interaction between the two vertices (qubits) a, b

U { a , b } = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 − 1 ]

And

| + ⟩ = | 0 ⟩ + | 1 ⟩ 2

Alternative definition

An alternative and equivalent definition is the following.

Define an operator K G ( a ) for each vertex a of G:

K G ( a ) = σ x ( a ) ∏ b ∈ N ( a ) σ z ( b )

where N(a) is the neighborhood of a (that is, the set of all b such that ( a , b ) ∈ E ) and σ x , y , z are the pauli matrices. The graph state | G ⟩ is then defined as the simultaneous eigenstate of the N = | V | operators { K G ( a ) } a ∈ V with eigenvalue 1:

K G ( a ) | G ⟩ = | G ⟩

References

Graph state Wikipedia
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