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Distributing Graph States Over Arbitrary Quantum Networks
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==Protocol Description== '''Graph Operations:''' * '''Vertex Deletion.''' Removes one vertex and all the associated edges from the graph. ** Implemented by the Pauli Measurement of the relevant qubit in the ''Z basis''. * '''Local Complementation.''' Inverts the subgraph induced by the neighborhood <math>N_a</math> of the concerned vertex <math>a</math>. ** Implemented by the quantum operator <math>U_a^\tau = e^{-i \frac{\pi}{4}X_a} \bigotimes_{b \in N_a} e^{i \frac{\pi}{4}Z_b}</math> acting on the qubits <math>a \cup N_a</math>. * '''Edge Addition.''' Add an edge between two nonadjacent vertices. ** Implemented by applying a ''controlled-Z'' between the desired nodes. * '''Edge Deletion.''' Delete an edge between two adjacent vertices. ** Implemented by applying a ''controlled-Z'' between the desired nodes. * '''Entanglement Swapping.''' ** Implemented by applying ''controlled-Z'' gate followed by two single qubit ''Y-measurement''. ===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. ===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.
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