Routing Entanglement in the Quantum Internet

This 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: Multi Party, Specific Task.

Assumptions

• Each network repeater node 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 network repeater node is aware of the overall network topology, as well as the locations of the $K$ Source-Destination pairs.

Outline

In 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

Quantum network with topology described by a graph $G(V,E)$:

• Each of the $N=|V|$ nodes is equipped with a quantum repeater.
• Each of the $M=|E|$ edges is a lossy optical channel of range $L_{i}$ (km) and power transmissivity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta_i \propto e^{-\alpha L_i}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i \in E} and $\alpha$ depends on the material of the channel.
• $K$ Source-Destination pairs $(A_{j},B_{j})$, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 \leq j \leq k} situated at (not necessarily distinct) nodes in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} .

Considering this graph$G$ we adopt the following notation for the repeater network.

• Each node Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v \in V} is a repeater.
• Each edge $e\in E$ is a physical link connecting two repeater nodes.
• $S(e)\in \mathbb {Z} ^{+}$ is an integer edge weight corresponding to the number of parallel channel across edge $e$.
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{N}(v)} is the set of neighbor edges of $v$
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(v) = |\mathcal{N}(v)|} is the degree of node Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} .
• $\sum _{e\in {\mathcal {N}}(v)}S(e)$ is the number of memories at node $v$.
• $d_{A}$ distance of a node to the node A, using the $\mathbb {L} ^{2}$ norm.
• An ebit represents a maximally entangled qubit.

Protocol Description

• All of the three protocols below assume that time is slotted and each memory can hold a qubit perfectly for $T\geq 1$ time slot. After this time the stored qubit completely decoheres.
• Each time slot $t$, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=1,2,... } is divided into 2 phases:
• External Phase:
  • Each of the $S(e)$ pairs of memories across and edge $e$ attempts to establish an EPR pair.

  - An entanglement attempt across any one of the $S(e)$ parallel links across edge $e$ succeeds with probability $p_{0}(e)\sim \eta (e)$, where $\eta (e)\sim e^{-\alpha L_{e}}$ is the transmissivity of a lossy optical channel of length $L(e)$.

  - The probability that one or more ebits are established across an edge $e$ is $p(e)=1-(1-p_{0})^{S(e)}$.

  - Assuming $S(e)=S,$ give us $p(e)=p,$ $\forall e\in E$.

  • Using two-way classical communication over edge $e(u,v)$, neighboring repeater nodes $u,v$ learn which of the S(e) parallel links (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 $q$.

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

The maximum number of ebits that can be shared between node $a$ and node $b$ after one time-slot is $min\{d(a),d(b)\}$, assuming $S$ 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 $K$.

Multipath Routing of a Single Entanglement Flow

Input: $K=1$, Source-Destination pair $(A_{1},B_{1})$.

Output: Quantum network with ebits shared between the end points of $(A_{1},B_{1})$.

Protocol for Entanglement Routing with Global Link-state Information

After the External Phase

1. $m=1$
2. $G_{m}$ = Subgraph induced by the successful external links and the repeater nodes after the external phase.
3. While (True):
   1. $S_{m}$ = shortest path in $G_{m}$ connecting $A_{1}$ with $B_{1}$.
   2. If $S_{m}$ is empty: Break While Loop.
   3. Else:
      1. Set $L_{m}$ as the length of $S_{m}$.
      2. Try connecting all internal links along the nodes of $S_{m}$
         //successfully generating an ebit between $A_{1}$ and $B_{1}$ with probability $q^{L_{m}-1}$
   4. $G_{m+1}$ = $G_{m}-$ all external and internal links of $S_{m}$.
   5. $m=m+1.$

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

1. For every repeater except $A_{1}$ and $B_{1}$:
   Let $u$ be the node of this iteration.
   1. If less than one of the neighboring external links is successful:
      no internal links are attempted since this repeater node can not be part of a path from $A_{1}$ to $B_{1}$.
   2. If two or more neighboring external links are successful:
      Let $v$ be the node linked to $u$ with the smallest $d_{A_{1}}$ and $w$ be the node linked to $u$ with the smallest $d_{B_{1}}$.
      Attempt a BSM on node $u$ on the memories connected to $v$ and $w$.
      1. If four neighboring external links are successful:
         Attempt a BSM on node $u$ on the remaining memories disconsidering the two memories from the previous step.

Protocol for Simultaneous Entanglement Flows with Link-state Information

Input: $K=2$, Source-Destination pairs $(A_{1},B_{1})$ and $(A_{2},B_{2})$.

Output: Quantum network with ebits shared between the end points of $(A_{1},B_{1})$ and $(A_{2},B_{2})$.

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 $\theta$ 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

• The goal of the optimal repeater strategy is to achieve:
  • Maximum entanglement generation rate for a single sender and receiver ($K=1$).

  - Above a (percolation) threshold determined by $p$ and $q$ the entanglement generation rate will depend only linearly on the transmissivity $\eta$ of a single link in the network.

• Maximum rate regions simultaneously achievable by the entanglement flows ($K\geq 2$).

  - Above a (percolation) threshold determined by $p$ and $q$ 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

• 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 $R_{1}$ reduces to the classical max-flow min-cut problem.
• Azuma and Kato (2016) looked at an "aggregated" protocol in which the repeater protocols run in parallel.
• 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.
• 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: Briegel et al. (1998), Jiand et al. (2008) and Muralidharan et al. (2016).

References

1. Pant et al. (2019)

*contributed by Lucas Arenstein