Graph state

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

${\left|G\right\rangle }=\prod _{(a,b)\in E}U^{\{a,b\}}{\left|+\right\rangle }^{\otimes V}$

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

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

And

${\left|+\right\rangle }={\frac {{\left|0\right\rangle }+{\left|1\right\rangle }}{\sqrt {2}}}$

Alternative definition

An alternative and equivalent definition is the following.

Define an operator $K_{G}^{(a)}$ for each vertex a of G:

$K_{G}^{(a)}=\sigma _{x}^{(a)}\prod _{b\in N(a)}\sigma _{z}^{(b)}$

where N(a) is the neighborhood of a (that is, the set of all b such that $(a,b)\in E$) and $\sigma _{x,y,z}$ are the pauli matrices. The graph state ${\left|G\right\rangle }$ is then defined as the simultaneous eigenstate of the $N=\left|V\right|$ operators $\left\{K_{G}^{(a)}\right\}_{a\in V}$ with eigenvalue 1:

$K_{G}^{(a)}{\left|G\right\rangle }={\left|G\right\rangle }$

See also

• Entanglement
• Cluster state

References

• M. Hein; J. Eisert; H. J. Briegel (2004). "Multiparty entanglement in graph states". Physical Review A. 69: 062311. doi:10.1103/PhysRevA.69.062311.
• S. Anders; H. J. Briegel (2006). "Fast simulation of stabilizer circuits using a graph-state representation". Physical Review A. 73: 022334. doi:10.1103/PhysRevA.73.022334.
• Graph states on arxiv.org