Editing Distributing Graph States Over Arbitrary Quantum Networks (section)

===Arbitrary Graph State Distribution Protocol===
To distribute an arbitrary graph state we first distribute the edge-decorated complete graph state. From this graph we can construct any other graph state by measuring each edge-qubit with a:
* ''Z measurement'' to delete this vertex and its edges
or a
* ''Y measurement'' to delete the vertex but keep the edges.

'''Input'''
* Arbitrary graph state to distribute
* Arbitrary set <math>W</math> of the <math>k</math> participating nodes.
'''Output'''
* Arbitrary graph state distributed over <math>W</math>.

'''Arbitrary Graph State Distribution Algorithm'''
# Solve the [https://en.wikipedia.org/wiki/Steiner_tree_problem Steiner Tree Problem] on the network for the <math>k</math> nodes.
#* Each one of the nodes of this tree has an central leaf <math>l_n</math>, <math>1 \leq n \leq k</math>.
# For <math>j=1</math> to <math>k</math>:
## Distribute a (k-1-j)-GHZ state on the node of <math>l_j</math> using the [[Distributing Graph States Over Arbitrary Quantum Networks#Protocol Description|GHZ-Distribution Algorithm ]].
## Delete vertices from the tree in order to have the covering tree for the set <math>W \setminus \{l_j\}</math>.
# For each one of the <math>k</math> nodes apply local operations to generate the edge-decorated graph state.
# For each one of the <math>k</math> nodes construct the desired arbitrary state by applying <math>Z</math> and <math>Y</math> measurements.