Editing Routing Entanglement in the Quantum Internet

<!-- This is a comment. You can erase them or write below -->
<!-- to do -->

<!-- Intro: brief description of the protocol -->
<!-- to do -->
This [https://www.nature.com/articles/s41534-019-0139-x example protocol (1)] implements the task of [[Entanglement Routing]] considering different scenarios. All of the protocols presented are applicable to quantum networks with arbitrary topology but their analysis is only concerned with the 2D grid network. They develop protocols for multipath routing of a single entanglement ﬂow with global or local state information of the other quantum repeater nodes and a protocol for multipath routing of simultaneous entanglement ﬂows with repeaters with local state information. 

'''Tags:'''  [[:Category: Multi Party Protocols|Multi Party]], [[:Category: Specific Task|Specific Task]].
<!-- I'm not sure if it is a Building Block or not [[:Category:Building Blocks|Building Block]] -->
<!--Tags: related pages or category -->
<!-- to do -->
==Assumptions==
<!-- It describes the setting in which the protocol will be successful. -->
* Each quantum repeater node is equipped with:
** Quantum memories that can hold a qubit perfectly for some predefined time.
** Entanglement sources.
** Ability to perform Bell state measurements between any pair of locally-held qubits.
** Classical computing resources and communication interface.
* Each quantum repeater node is aware of the overall network topology, as well as the locations of the <math>K</math> Source-Destination pairs.

==Outline==
<!-- A non-mathematical detailed outline which provides a rough idea of the concerned protocol -->
In [https://www.nature.com/articles/s41534-019-0139-x.pdf (1)] they develop and analyze routing protocols for generating maximally entangled qubits (ebits) simultaneously between single or multiple pairs of senders and receivers in a quantum network by exploiting multiple paths in the network. They introduce protocols for quantum repeater nodes in the following scenarios: 
* Multipath routing of a '''single''' entanglement ﬂow:
** Considering nodes with global link-state information (the state of every link is known to every repeater in the network and can be used).
** Considering nodes with local link-state information (a link knows the states of it neighbors).
* Multipath routing of '''simultaneous''' entanglement ﬂows:
** Considering nodes with local link-state information.
All of the protocols above considering their characteristics are divided in two phases with the objective of creating an unbroken end-to-end connection between the source and destination nodes. The external and the internal phase which occur in this order.
*External phase: each pair of memories (neighboring repeaters) across an edge attempts to establish a shared entangled (EPR) pair.
*Internal phase: BSMs are attempted within each memory (repeater node) based on the successes and failures of the neighboring links in the external phase.

The high-level objective is: Considering the assumptions of quantum and classical operations at each of the repeater nodes of the underlying network, what operations should be performed at the repeater to achieve maximum entanglement-generation rate for one sender and receiver or achieve maximum rate region for multiple ﬂows of entanglement.

==Notation==
<!--  Connects the non-mathematical outline with further sections. -->
Quantum network with topology described by a graph <math>G(V,E)</math>:
* Each of the <math>N=|V|</math> nodes is equipped with a quantum repeater.
* Each of the <math>M=|E|</math> edges is a lossy optical channel of range <math>L_i</math> (km) and power transmissivity <math>\eta_i \propto e^{-\alpha L_i}</math>, <math>i \in E</math> and <math>\alpha</math> depends on the material of the channel.
* <math>K</math> Source-Destination pairs <math>(A_j, B_j)</math>, <math>1 \leq j \leq k</math> situated at (not necessarily distinct) nodes in <math>V</math>.
Considering this graph<math>G</math> we adopt the following notation for the repeater network.
* Each node <math>v \in V</math> is a repeater. 
* Each edge <math>e \in E</math> is a physical link connecting two repeater nodes.
* <math>S(e) \in \mathbb{Z}^+</math> is an integer edge weight corresponding to the number of parallel channel across edge <math>e</math>.
* <math>\mathcal{N}(v)</math> is the set of neighbor edges of <math>v</math>
* <math>d(v) = |\mathcal{N}(v)|</math> is the degree of node <math>v</math>.
* <math>\sum_{e \in \mathcal{N}(v)} S(e)</math> is the number of memories at node <math>v</math>.
* <math>d_A</math> distance of a node to the node A, using the <math>\mathbb{L}^2</math> norm.
* An ebit represents a maximally entangled qubit.

<!-- ==Knowledge Graph== -->
<!-- Add this part if the protocol is already in the graph -->
<!--{{graph}} -->
==Protocol Description==
<!-- Mathematical step-wise protocol algorithm helpful to write a subroutine. -->

* All of the three protocols below assume that time is slotted and each memory can hold a qubit perfectly for <math>T \geq 1</math> time slot before the stored qubit completely decoheres.
* Each time slot <math>t</math>, <math>t=1,2,... </math> is divided into 2 phases:
* '''External Phase:'''
** Each of the <math>S(e)</math> pairs of memories across and edge <math>e</math> attempts to establish an EPR pair.
*:- An entanglement attempt across any one of the <math>S(e)</math> parallel links across edge <math>e</math> succeeds with probability <math>p_0(e) \sim \eta(e)</math>, where <math>\eta(e) \sim e^{-\alpha L_e}</math> <!--is the transmissivity of a lossy optical channel of length <math>L(e)</math>.-->
*:- The probability that one or more ebits are established across an edge <math>e</math> is <math>p(e)=1-(1-p_0)^{S(e)}</math>.
*:- Assuming <math>S(e)=S,</math> give us <math>p(e)=p,</math> <math>\forall e \in E</math>.
** Using two-way classical communication over an edge <!--<math>e(u, v)</math>-->, neighboring repeater nodes <!--<math>u, v</math>--> learn which of the S(e) (if any) succeeded in the external phase, in a given time slot.

* '''Internal Phase:'''
** BSMs are attempted locally at each repeater node between pairs of qubit memories based on the success and failure of the neighboring links in the external phase.
*:- These BSM attempts are called internal links, i.e., links between memories internal to a repeater node.
*:- Each of these internal-link attempts succeed with probability <math>q</math>.

At the end of one time-slot a along a path comprising of <math>k</math> edges (and thus <math>(k-1)</math> repeater nodes) one ebit is successfully shared between the end points of the path with probability <math>p^k q^{k-1}</math>.

The maximum number of ebits that can be shared between node <math>a</math> and node <math>b</math> after one time-slot is <math>min\{d(a), d(b)\}</math>, assuming <math>S</math> is the same over all edges.

The protocols described below focus on finding the optimal strategy for each repeater node in order to decide which locally held qubits to attempt BSMs during the internal phase of a
time slot. Based on the outcomes of the external phase and considering global or local link-state knowledge and <math>K</math>.

===Multipath Routing of a Single Entanglement Flow===
'''Input:''' <math>K=1</math>, Source-Destination pair <math>(A_1, B_1)</math>.

'''Output:''' Quantum network with ebits shared between the end points of <math>(A_1, B_1)</math>.

====Protocol for Entanglement Routing with Global Link-state Information====
After the External Phase
# <math>m = 1</math>
# <math>G_m</math> = Subgraph induced by the successful external links and the repeater nodes after the external phase.
# While (True):
## <math>S_m</math> = shortest path in <math>G_m</math> connecting <math>A_1</math> with <math>B_1</math>. 
## If <math>S_m</math> is empty: Break While Loop.
## Else:
### Set <math>L_m</math> as the length of <math>S_m</math>.
### Try connecting all internal links along the nodes of <math>S_m</math> <br />//successfully generating an ebit between <math>A_1</math> and <math>B_1</math> with probability <math>q^{L_m - 1}</math>
## <math>G_{m+1}</math> = <math>G_m -</math> all external and internal links of <math>S_m</math>.
## <math>m = m + 1.</math> 

====Protocol for Entanglement Routing with Local Link-state Information====

Knowledge of success and failure of the External Phase is communicated only to the two repeater nodes connected by the link.
Repeater nodes need to decide on which pair(s) of memories BSMs should be attempted (which internal links to attempt), based only on information about the states of external links adjacent to them.

After the External Phase
# For every repeater except <math>A_1</math> and <math>B_1</math>: <br /> Let <math>u</math> be the node of this iteration.
## If less than one of the neighboring external links is successful: <br />Then no internal links are attempted since this repeater node can not be part of a path from <math>A_1</math> to <math>B_1</math>.
## Else If two or more neighboring external links are successful: <br />Then let <math>v</math> be the node linked to <math>u</math> with the smallest <math>d_{A_1}</math> and <math>w</math> be the node linked to <math>u</math> with the smallest <math>d_{B_1}</math>. <br /> Attempt a BSM on node <math>u</math> on the memories connected to <math>v</math> and <math>w</math>.
### Else If four neighboring external links are successful: <br /> Then attempt a BSM on node <math>u</math> on the remaining memories disconsidering the two memories from the previous step. 

===Protocol for Simultaneous Entanglement Flows with Link-state Information===
'''Input:''' <math>K=2</math>, Source-Destination pairs <math>(A_1, B_1)</math> and <math>(A_2, B_2)</math>.

'''Output:''' Quantum network with ebits shared between the end points of <math>(A_1, B_1)</math> and <math>(A_2, B_2)</math>.

This situation motivates the '''Multi-ﬂow Spatial-division''' rule which divides the the network between two spatial regions corresponding to the two flows.

* When the shortest path connecting the two source-destination pairs '''do not''' cross the network is divided between two spatial regions corresponding to the two ﬂows.
** For each one of this regions we apply the Protocol for Entanglement Routing with Local Link-state Information.

* When the shortest path connecting the two source-destination pairs '''do''' cross the network is divided between two spatial regions corresponding to the two ﬂows.
** Considering the square-grid topology the two spatial regions are divided by two crossing lines with an angle <math>\theta</math> between them (forming a hourglass shape).
** For each one of this regions we apply the Protocol for Entanglement Routing with Local Link-state Information.

==Properties==
<!-- important information on the protocol: parameters (threshold values), security claim, success probability... -->
<!-- to do -->
* The goal of the optimal repeater strategy is to achieve:
** Maximum entanglement generation rate for a single sender and receiver (<math>K=1</math>).
*:- Above a (percolation) threshold determined by <math>p</math> and <math>q</math> the entanglement generation rate will depend only linearly on the transmissivity <math>\eta</math> of a single link in the network.
* Maximum rate regions simultaneously achievable by the entanglement flows (<math>K \geq 2</math>).
*:- Above a (percolation) threshold determined by <math>p</math> and <math>q</math> this protocol signiﬁcantly exceeds the ones that each repeater node makes BSM decisions by simply time-sharing between catering to the individual ﬂows.

==Further Information==
<!-- theoretical and experimental papers including requirements, security proof (important), which protocol does it implement, benchmark values... -->
<!-- to do -->
* [https://arxiv.org/abs/1601.00966 Pirandola (2016)] analyzed entanglement-generation capacities of repeater networks assuming ideal repeater nodes and argued that for a single ﬂow the maximum entanglement-generation rate <math>R_1</math> reduces to the classical max-ﬂow min-cut problem.
* [https://arxiv.org/abs/1606.00135 Azuma and Kato (2016)] looked at an "aggregated" protocol in which the repeater protocols run in parallel.
* [https://arxiv.org/abs/1610.05238 Schoute et al. (2016)] developed routing protocols on speciﬁc network topologies and found scaling laws as functions on the number of qubits in the memories at nodes, and the time and space consumed by the routing algorithms, under lossless and noiseless assumptions.
* [https://arxiv.org/abs/quant-ph/0612167 Acín et al. (2006)] have considered the problem of entanglement percolation where neighboring nodes share a perfect lossless pure state.
There is an extensive literature of quantum networks analyzing repeaters in a linear chain for example:
[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.81.5932 Briegel et al. (1998)], [https://arxiv.org/abs/0809.3629 Jiand et al. (2008)] and [https://www.nature.com/articles/srep20463 Muralidharan et al. (2016)].
==References==
<!-- to do -->
# [https://www.nature.com/articles/s41534-019-0139-x Pant et al. (2019)]

<!-- Version 1 -->
<div style='text-align: right;'>''Contributed by Lucas Arenstein during the QOSF Mentorship Program''</div>
<div style='text-align: right;'>''Mentor: Shraddha Singh</div>