Asynchronous Entanglement Routing

• Asynchronous Entanglement Routing protocols are decentralized quantum network methods that use local, event-driven algorithms to generate, maintain, and swap Bell pairs without global synchronization.
• They employ dynamic logical topologies such as DODAGs and distributed spanning trees, utilizing local link-state updates to optimize end-to-end entanglement rates and resource use.
• These protocols demonstrate significant advantages in scalability, rate optimization, and robustness over traditional synchronous methods in heterogeneous quantum repeater networks.

Asynchronous Entanglement Routing is a class of quantum network protocols that replace globally synchronized, time-slot-based mechanisms with local, event-driven, and distributed algorithms for generating, maintaining, and consuming entanglement links. Such protocols are motivated by the fundamental constraints of quantum hardware, including probabilistic entanglement generation, limited coherence times, and restrictions on classical signaling. Instead of requiring a global view or coordination, asynchronous entanglement routing algorithms use only local link-state information, maintain a dynamically evolving logical topology (e.g., trees or directed acyclic graphs) atop the physical network, and optimize end-to-end entanglement throughput by judicious path selection, link preservation, and swap operation scheduling. These approaches have demonstrated substantial advantages in rate, scalability, and resource utilization compared to traditional synchronous techniques, especially in large or heterogeneous quantum networks (Yang et al., 2023, Yang et al., 12 Aug 2024, Pant et al., 2017, Pouryousef et al., 3 May 2024, Milligen et al., 2023, Chakraborty et al., 2019, Li et al., 2020).

1. Network Model and Core Assumptions

The canonical setting is a quantum repeater network modeled as an undirected graph $G = (V, E)$, where each node $v \in V$ is a quantum repeater and each edge $e \in E$ is an optical link supporting probabilistic Bell pair generation. Each link attempt succeeds independently with probability $p(e)$, often assumed homogeneous, $p(e) = p$. Node qubit storage is limited, and each link or storage location is subject to a coherence time $T_{co}$: a successfully generated Bell pair decoheres and is unusable after $T_{co}$ (measured in discrete slots or as a continuous parameter).

At the logical level, at any moment the "instant topology" $G' = (V', E')$ comprises only those physical links for which entanglement exists and remains within its coherence window. Swapping involves Bell-state measurements at repeaters, with success probability $q$, to concatenate adjacent Bell pairs. For many analyses, edge and node capacities are normalized, and purification overheads are absorbed into effective link rates (Yang et al., 2023).

2. Dynamic Topology Management

Asynchronous routing protocols rely on distributed, local updates to a logical network overlay representing available entanglement links. Principal structures include:

• Destination-Oriented Directed Acyclic Graphs (DODAGs): Each node maintains a "rank" ($\rho(v)$) relative to a chosen root, updating this rank according to the minimal cost of reaching the root via existing entanglement links. Nodes broadcast status messages (e.g., DIO, DAO, DIS), join the DODAG after confirming entanglement on an adjacent edge, and dynamically recompute ranks upon link addition or removal. Graph updates react to local events; there is no global slot or synchronization (Yang et al., 2023, Yang et al., 12 Aug 2024).
• Distributed Spanning Trees: Variants built via local message flooding or distributed algorithms (such as Gallagher–Humblet–Spira) organize nodes into tree structures rooted at key nodes. Maintenance involves only neighbor-to-neighbor interactions and is robust to local link failures (Yang et al., 2023, Yang et al., 12 Aug 2024).
• Forest/Multitree Extensions: Large or topologically clustered networks use simultaneously maintained DODAGs (a "forest") to reduce route stretch and balance resource usage. Nodes can join multiple trees under explicit rules to prevent cycles and redundant routes. This strategy increases entanglement rate, particularly in regular grids or clusters (Yang et al., 12 Aug 2024).

Each logical link tracks a timestamp; links are deleted and topologies updated when entanglements expire or are consumed by swaps. Only those links lying on the selected route for a request are consumed; all others are preserved for future requests until decoherence (Yang et al., 2023).

3. Asynchronous Routing Algorithms

Event-driven operation replaces deterministic slotting: nodes respond to two primary types of events—local logical graph updates and external connection requests.

• Connection Request Handling: Upon a request to establish an entangled pair between $s$ and $t$, the source queries its local data structure (tree/DODAG/forwarding table), determines the next-hop neighbor (based on minimal rank or tree pointer), and initiates a Bell-state measurement with the qubit entangled to the previous node. If the swap fails, topology is refreshed, and the operation is retried. The process recursively proceeds until the destination is reached and a Bell pair is achieved (Yang et al., 2023, Yang et al., 12 Aug 2024).
• Algorithmic Variants:
  • Local-metric policies (distance or tree rank) for path advancement.
  • Disjoint-path routing for parallel entanglement flows (especially in multi-user scenarios) (Pant et al., 2017).
  • Local greedy or "best effort" routing rules based solely on immediate neighbor link states ("asynchronous local link-state routing") (Pant et al., 2017, Chakraborty et al., 2019).
  • Time-multiplexed schemes, in which repeaters buffer several attempted links before performing swaps, with all decisions based on locally available memory states (Milligen et al., 2023).

No global schedule or slot boundary is needed, except possibly for resetting expired or decohered qubits. Asynchronous operation enables requests to be initiated, routed, and completed at arbitrary times, as supported by capacity-aware scheduling algorithms (Li et al., 2020).

4. Rate Optimization and Path Selection

The objective is to maximize the end-to-end entanglement rate $\xi$ (or $R$), i.e., the long-term average number of delivered Bell pairs per time slot or per physical time. Two paradigmatic rate formulas are as follows:

• Synchronous global-slot protocols:

$\xi_{syn}(s,t) = p^{l_{s,t}} q^{l_{s,t}-1} n_{s,t}$

where $l_{s,t}$ is the shortest (mean) path length and $n_{s,t}$ the number of disjoint concurrent paths (≤4 in a 2D grid).

• Asynchronous DODAG protocol:

$$\xi_{DODAG}(s,t) = p^{l_{s,r} + l_{r,t} - (l'_{s,r} + l'_{r,t})} q^{l_{s,r} + l_{r,t} - 1} n_{s,t}$$

Here, $r$ is the root, $l_{s,r}$, $l_{r,t}$ are the hop-paths, $l'_{s,r}$, $l'_{r,t}$ count present Bell pairs, and $n_{s,t}$ is path multiplicity (Yang et al., 2023). Asynchronous rates exploit pre-existing direct entanglement, scale with memory time $T_{co}$, and allow multiple flows to concurrently utilize entanglement residuals.

Multipath and local-knowledge policies further reduce exponential decay with distance, compared to single-path chains (Pant et al., 2017, Milligen et al., 2023). Optimization also considers routing fairness, resource allocation (proportional sharing, progressive filling, or propagatory update methods), and extension to streaming or continuous-request settings (Li et al., 2020).

5. Performance Analysis and Comparative Studies

Asynchronous schemes exhibit multiple theoretical and practical advantages:

• End-to-End Rate Gains: Asynchronous protocols increase the rate upper bound and offer rates that strictly improve with coherence time. Analytical results and simulations on 2D grids (e.g., 26×26) show that asynchronous protocols achieve significantly higher rates at all distances and decay more slowly with distance compared to synchronous protocols. The advantage persists even for short distances, as local routing bypasses global root bottlenecks and exploits opportunistic, residual Bell pairs (Yang et al., 2023, Yang et al., 12 Aug 2024, Pant et al., 2017).
• Scalability: As network size increases, synchronous protocols require global slot synchronization and memory times scaling with network diameter; asynchronous methods require only local updates and coherence proportional to local path latencies, yielding better scalability and feasibility for deployment in realistic, large networks (Yang et al., 2023, Yang et al., 12 Aug 2024).
• Efficiency across Topologies: Multi-tree schemes substantially improve rates (e.g., 30% in large grids over single-tree, 15–40% in real testbeds such as SURFnet and ESnet), especially when roots are chosen to minimize eccentricity or cover natural clusters (Yang et al., 12 Aug 2024). Linear chain and barbell topologies show weaker gains due to inherent structure.
• Cutoff and Memory Effects: Sequential and parallel minimal asynchronous protocols, including memory cutoff strategies (discarding qubits with excessive idle time), achieve optimized rates and fidelities for entanglement distribution and QKD. Sequential protocols have comparable or superior performance to parallel for inhomogeneous or dynamically routed networks, with simpler messaging requirements (Pouryousef et al., 3 May 2024).
• Multipath, Multiuser, and Load Sharing: Local spatial division of flows supports simultaneous multiuser entanglement generation, beating classical time-division and enabling robust concurrent operation (Pant et al., 2017, Chakraborty et al., 2019, Li et al., 2020).

6. Advanced Protocol Features and Design Guidelines

Several advanced protocol features, extensions, and operational recommendations are established in the literature:

• Local Information Sufficiency: Policies leveraging only neighbor link-state are sufficient to extract most of the multipath gain without global link-state flooding, significantly reducing control plane overhead (Pant et al., 2017).
• Time-Multiplexing: Block-based schemes (parameterized by block length $k$) allow tradeoff between success probability (multi-attempt advantage) and the risk of decoherence. An optimal $k_{\rm opt}$ balances latency, link probability, and memory lifetime (Milligen et al., 2023).
• Asynchronous Scheduling Algorithms: Proportional share, progressive filling, and propagatory update allocation heuristics (Editor’s term) can be run in an event-driven, asynchronous fashion on each network update. These algorithms ensure fairness, high utilization, and rapid adaptation to request arrivals and topology changes (Li et al., 2020).
• Robustness to Topology Dynamics: DODAG and multitree forests immediately respond to failures and repairs via local messages, supporting dynamic networks, partial outages, and evolving traffic patterns (Yang et al., 12 Aug 2024, Yang et al., 2023).
• Guidelines:
  • Use sequential (hop-by-hop) routing for inhomogeneous or dynamically changing networks; parallel when the coherence time is high and delays are uniform (Pouryousef et al., 3 May 2024).
  • Employ cutoff strategies for local memories to prevent entanglement quality degradation at mild rate expense.
  • Choose root/treetop placements to minimize path-lengths for dominant flows.
  • Plan repeater spacing so that per-link delays are less than memory coherence for the intended traffic (Pouryousef et al., 3 May 2024, Milligen et al., 2023).
  • For large or clustered networks, prefer multi-tree forests to single-tree for improved rates and resilience (Yang et al., 12 Aug 2024).

7. Practical Implementation, Applications, and Outlook

Asynchronous entanglement routing protocols are well suited for NISQ-era and near-future quantum networks where quantum memories are limited, and classical communication is non-negligible. Their decentralization, scalability, and ability to preserve unused resources make them directly implementable on testbed platforms like SURFnet, ESnet, and Internet2 (Yang et al., 12 Aug 2024, Pouryousef et al., 3 May 2024). They provide a practical foundation for distributed quantum key distribution, multi-user quantum networking, and quantum internet services, establishing a versatile, efficient, and hardware-adapted routing layer. As hardware advances improve entanglement generation rates and memory lifetimes, the performance and impact of asynchronous protocols will become increasingly pronounced (Yang et al., 2023, Yang et al., 12 Aug 2024, Milligen et al., 2023).