# 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 = (VE), with the set of vertices V and the set of edges E, the corresponding graph state is defined as

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

where ${\displaystyle {\left|+\right\rangle }=({\left|0\right\rangle }+{\left|1\right\rangle })/{\sqrt {2}}}$ and the operator ${\displaystyle U^{\{a,b\}}}$ is the controlled-Z interaction between the two vertices (qubits) a, b

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

### Alternative definition

An alternative and equivalent definition is the following.

Define an operator ${\displaystyle S_{v}}$ for each vertex v of G:

${\displaystyle S_{v}=\sigma _{x}^{(v)}\prod _{u\in N(v)}\sigma _{z}^{(u)}}$

where ${\displaystyle \sigma _{x,y,z}}$ are the Pauli matrices and N(v) is the set of vertices adjacent to v. The ${\displaystyle S_{v}}$ operators commute. The graph state ${\displaystyle {\left|G\right\rangle }}$ is defined as the simultaneous ${\displaystyle +1}$-eigenvalue eigenstate of the ${\displaystyle \left|V\right|}$ operators ${\displaystyle \left\{S_{v}\right\}_{v\in V}}$:

${\displaystyle S_{v}{\left|G\right\rangle }={\left|G\right\rangle }}$

## Examples

• If ${\displaystyle G=P_{3}}$ is a three-vertex path, then the ${\displaystyle S_{v}}$ stabilizers are
{\displaystyle {\begin{aligned}\sigma _{x}\otimes \sigma _{z}\otimes I,\\\sigma _{z}\otimes \sigma _{x}\otimes \sigma _{z},\\I\otimes \sigma _{z}\otimes \sigma _{x}\end{aligned}}}

The corresponding quantum state is

${\displaystyle {\left|P_{3}\right\rangle }={\frac {1}{\sqrt {8}}}({\left|000\right\rangle }+{\left|100\right\rangle }+{\left|010\right\rangle }-{\left|110\right\rangle }+{\left|001\right\rangle }+{\left|101\right\rangle }-{\left|011\right\rangle }+{\left|111\right\rangle })}$
• If ${\displaystyle G=K_{3}}$ is a triangle on three vertices, then the ${\displaystyle S_{v}}$ stabilizers are
{\displaystyle {\begin{aligned}\sigma _{x}\otimes \sigma _{z}\otimes \sigma _{z},\\\sigma _{z}\otimes \sigma _{x}\otimes \sigma _{z},\\\sigma _{z}\otimes \sigma _{z}\otimes \sigma _{x}\end{aligned}}}

The corresponding quantum state is

${\displaystyle {\left|K_{3}\right\rangle }={\frac {1}{\sqrt {8}}}({\left|000\right\rangle }+{\left|100\right\rangle }+{\left|010\right\rangle }-{\left|110\right\rangle }+{\left|001\right\rangle }-{\left|101\right\rangle }-{\left|011\right\rangle }-{\left|111\right\rangle })}$

Observe that ${\displaystyle {\left|P_{3}\right\rangle }}$ and ${\displaystyle {\left|K_{3}\right\rangle }}$ are locally equivalent to each other, i.e., can be mapped to each other by applying one-qubit unitaries. Indeed, switching ${\displaystyle \sigma _{x}}$ and ${\displaystyle \sigma _{y}}$ on the first and last qubits, while switching ${\displaystyle \sigma _{y}}$ and ${\displaystyle \sigma _{z}}$ on the middle qubit, maps the stabilizer group of one into that of the other.

More generally, two graph states are locally equivalent if and only if the corresponding graphs are related by a sequence of so-called local complementation" steps, as shown by Van den Nest et al. (2005).