Routing Entanglement in the Quantum Internet: Difference between revisions
Routing Entanglement in the Quantum Internet (edit)
Revision as of 20:03, 6 January 2022
, 6 January 2022Edit by Lucas: Corrections and suggestions from our last talk - Removed Kaushik's name
(Updated the terminology to match/agree with the newest protocol page I'm doing (I will finish the new one in the next couple of days)) |
(Edit by Lucas: Corrections and suggestions from our last talk - Removed Kaushik's name) |
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<!-- Mathematical step-wise protocol algorithm helpful to write a subroutine. --> | <!-- 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 | * 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: | * Each time slot <math>t</math>, <math>t=1,2,... </math> is divided into 2 phases: | ||
* '''External Phase:''' | * '''External Phase:''' | ||
** Each of the <math>S(e)</math> pairs of memories across and edge <math>e</math> attempts to establish an EPR pair. | ** 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>. | *:- 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>. | *:- 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>. | *:- 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 edge <math>e(u, v)</math>, neighboring repeater nodes <math>u, v</math> learn which of the S(e) | ** 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:''' | * '''Internal Phase:''' | ||
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*:- These BSM attempts are called internal links, i.e., links between memories internal to a repeater node. | *:- 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>. | *:- 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>. | 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>. | ||
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## <math>G_{m+1}</math> = <math>G_m -</math> all external and internal links of <math>S_m</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> | ## <math>m = m + 1.</math> | ||
====Protocol for Entanglement Routing with Local Link-state Information==== | ====Protocol for Entanglement Routing with Local Link-state Information==== | ||
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After the External Phase | 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. | # 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 />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>. | ## 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>. | ||
## If two or more neighboring external links are successful: <br /> | ## 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>. | ||
### If four neighboring external links are successful: <br /> | ### 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=== | ===Protocol for Simultaneous Entanglement Flows with Link-state Information=== | ||
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# [https://www.nature.com/articles/s41534-019-0139-x Pant et al. (2019)] | # [https://www.nature.com/articles/s41534-019-0139-x Pant et al. (2019)] | ||
<div style='text-align: right;'>'' | <!-- 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> | |||