Editing Distributing Graph States Over Arbitrary Quantum Networks (section)

===GHZ State Distribution Protocol===
'''Input'''
* N-GHZ state: <math>|\text{N-GHZ}\rangle = (|0\rangle^{\otimes N}+|1\rangle^{\otimes N})/\sqrt{2}</math>.
* That is locally equivalent to: <math>(|0\rangle|+\rangle^{\otimes N-1}+|1\rangle|-\rangle^{\otimes N-1})/\sqrt{2}</math> that is called a star graph '''upload image star_graph_distributing.svg here'''

<!--[[File:star_graph_distributing.svg]]-->

* Arbitrary set <math>W</math> of the network nodes.
** Assuming that the network topology is already given to us.

'''Output'''
* N-GHZ state distributed over <math>W</math>.

'''GHZ-Distribution Algorithm'''
# Find a minimal tree covering all the nodes of <math>W</math> 
# <math>l</math> is a random leaf from the tree of step 1.
# For <math>j=1</math> to <math>|W| - 1</math>:    
## Let <math>A</math> be any unprocessed node of <math>W</math> such as <math>l \notin A</math>.
## Apply the Start Expansion Algorithm on node <math>A</math> with <math>l</math> as <math>b</math> the center of the star.

'''Start Expansion Algorithm'''
This routine uses the Bell pairs of the node <math>A</math> to add the edges <math>(b,c_i)</math> to the graph state, as well as the edge <math>(b, a_0)</math> ''iff'' <math>A \in W</math>. 

# All the qubits <math>a_i, i \geq 0</math> of <math>A</math> are linked using Controlled-Z operations between all possible pairs.
# Local complementation is applied to the qubit <math>a_0</math> linked to <math>b</math>.
# If <math>A \notin W</math>:
## Remove <math>a_0</math> and all the edges within <math>A</math> by <math>Z</math>-measuring it
## Else (when <math>A \in W</math>):
###Keep <math>a_0</math> and apply  Controlled-Z gates to remove all edges within <math>A</math>.
#Apply a <math>Y</math>-measurement on all the other qubits <math>a_i \in A</math>, <math>i>0</math> creating the desired Start Graph.