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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.

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 ⟩ = \prod_{(a, b) \in E} U^{{a, b}} {| + ⟩}^{\otimes V}

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

U^{{a, b}} = [\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - 1 \end{array}]

And

| + ⟩ = \frac{| 0 ⟩ + | 1 ⟩}{\sqrt{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)} \prod_{b \in N (a)} σ_{z}^{(b)}

where N(a) is the neighborhood of a (that is, the set of all b such that $(a, b) \in E$ ) 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 ${K_{G}^{(a)}}_{a \in V}$ with eigenvalue 1:

K_{G}^{(a)} | G ⟩ = | G ⟩

References

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A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

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Graph states on arxiv.org

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Formal definition

Alternative definition

See also

References

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