Distributing Graph States Over Arbitrary Quantum Networks: Difference between revisions

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Added the properties part. I'm going to send the images by email/slack
(First version of this page. I still need to add the introduction, properties, images and make another good review of everything.)
(Added the properties part. I'm going to send the images by email/slack)
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In this protocol a quantum network is represented as a graph
In this protocol a quantum network is represented as a graph
''upload image here''
'''upload image qn_distributing.svg here'''
<!--[[File:.png]]-->
<!--[[File:qn_distributing.svg]]-->


The physical distribution of graph states are represented as graph operations ignoring
The physical distribution of graph states are represented as graph operations ignoring
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* <math>|A|</math> number of nodes of A.
* <math>|A|</math> number of nodes of A.
* <math>a_i \in A</math>, <math>i > 0</math> Represents each of the qubits of <math>A</math> that constitutes a Bell pair with a qubit <math>c_i</math> in another node of the network.
* <math>a_i \in A</math>, <math>i > 0</math> Represents each of the qubits of <math>A</math> that constitutes a Bell pair with a qubit <math>c_i</math> in another node of the network.
==Properties==
* The optimality of the protocol depends on the network topology and the wanted graph state to distribute.
* Analysing the consumption of Bell pair between nodes and considering the worst case scenario that is to entangle each qubit with the opposite one over a line network we have the following upper bounds:
{| class="wikitable"
|-
! Distribute
! Cost
! Bound
|-
|rowspan="2"| N-GHZ
| EPR
| <math>N-1</math>
|-
| T
| <math>1</math>
|-
|rowspan="2"| Arbitrary Graph State
| EPR
| <math>\left\lfloor \frac{N}{2} \right\rfloor^2</math>
|-
| T
| <math>\left\lfloor \frac{N}{2} \right\rfloor</math>
|-
|}


==Protocol Description==
==Protocol Description==
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'''Input'''
'''Input'''
* N-GHZ state: <math>|\text{N-GHZ}\rangle = (|0\rangle^{\otimes N}+|1\rangle^{\otimes N})/\sqrt{2}</math>.
* 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 '''link to star graph'''
* 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'''


<!-- Upload Star Graph Image-->
<!--[[File:star_graph_distributing.svg]]-->


* Arbitrary set <math>W</math> of the network nodes.
* Arbitrary set <math>W</math> of the network nodes.
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