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Path Selection for Quantum Repeater Networks
Rodney Van Meter, Takahiko Satoh, Thaddeus D. Ladd, William J. Munro, Kae Nemoto
TL;DR
Quantum repeater networks need practical path selection for heterogeneous topologies, but their quantum-specific behavior makes classical resource costs inadequate. The paper adapts Dijkstra’s algorithm using inverse link throughput as cost and evaluates it with detailed physical and classical-message simulations. Across heterogeneous paths, the calculated cost closely tracks total work and throughput, supporting Dijkstra’s use in this setting.
Problem
Heterogeneous quantum repeater networks require path selection, but quantum-state fragility, limited buffering, and memory decay make classical routing costs inadequate.
Method
The paper applies Dijkstra’s algorithm with inverse link throughput, measured in Bell pairs per second at a specified fidelity, as the link cost.
Results
0.88 or better coefficient of determination was obtained between calculated path cost and total work across two heterogeneous-path simulation sets.
Takeaways & Limitations
Dijkstra/BellGenT provides a readily calculated path cost that correlates strongly with actual throughput and total work, supporting its use for these quantum repeater networks.
Takeaways & Limitations
The simulations use basic qubus links limited to about 30 km per hop over high-quality telecom fiber, with other link types left for future work.
Abstract
from arXiv · showhide
Quantum networks will support long-distance quantum key distribution (QKD) and distributed quantum computation, and are an active area of both experimental and theoretical research. Here, we present an analysis of topologically complex networks of quantum repeaters composed of heterogeneous links. Quantum networks have fundamental behavioral differences from classical networks; the delicacy of quantum states makes a practical path selection algorithm imperative, but classical notions of resource utilization are not directly applicable, rendering known path selection mechanisms inadequate. To adapt Dijkstra's algorithm for quantum repeater networks that generate entangled Bell pairs, we quantify the key differences and define a link cost metric, seconds per Bell pair of a particular fidelity, where a single Bell pair is the resource consumed to perform one quantum teleportation. Simulations that include both the physical interactions and the extensive classical messaging confirm that Dijkstra's algorithm works well in a quantum context. Simulating about three hundred heterogeneous paths, comparing our path cost and the total work along the path gives a coefficient of determination of 0.88 or better.
1 Introduction
The paper addresses path selection in heterogeneous quantum repeater networks, where quantum-specific behavior makes classical routing costs inadequate. It adapts Dijkstra’s algorithm with a throughput-based link cost and evaluates the approach through detailed simulations.
- Motivation: Heterogeneous quantum networks require path selection for irregular topologies and links with varying length and quality.Prior work largely focused on linear repeater chains with identical links, while path selection affects connection performance and global network load.
- Motivation: Quantum path selection differs from classical routing because fragile quantum information, limited buffering, costly memory, and rapidly decaying fidelity constrain communication.These properties require acknowledged link-layer operations and introduce real-time communication demands.
- Approach: The proposed method applies Dijkstra’s algorithm to rank candidate paths in quantum repeater networks and evaluates throughput and global work.The algorithm is assessed against detailed simulations of physical interactions and classical messaging.
- Approach: The proposed link cost is the inverse of link throughput, measured in Bell pairs per second at a particular fidelity.Lower-level metrics such as laser pulses and quantum measurements estimate total path work but poorly prioritize links because they omit system factors affecting end-to-end performance.
- Evaluation: 0.88 or better coefficient of determination links calculated path cost with total work across two simulation sets of heterogeneous paths.The simulations include 46 paths varying from one to nine hops and 256 four-hop link combinations.
- Evaluation: More than 80% of tested path pairs selected the lower-cost path as the higher-throughput path.The results also indicate that throughput is limited by the bottleneck link, while total work depends on path length and all link qualities.
2 Difficulties: Differences Between Quantum and Classical Networks
Quantum networks differ from classical networks because quantum states cannot be copied, decay over time, and require probabilistic operations and classical coordination. Entanglement swapping and purification extend and improve Bell pairs, but consume resources and can reduce throughput.
- Quantum-specific constraints: Quantum information cannot be independently copied, so lost data cannot simply be retransmitted without risking the state.Quantum networks therefore use generic entangled states to teleport data.
- Quantum-specific constraints: Quantum memory lifetimes can be shorter than round-trip latencies, making storage duration a central communication constraint.Maintaining quantum states may require expensive hardware and strong error correction.
- Entanglement operations: Entanglement swapping combines two short-distance Bell pairs into one longer-distance pair, with each carefully managed step able to double its span.The operation requires classical information to be sent to both ends of the new entangled connection.
- Entanglement operations: Purification converts two imperfect Bell pairs into one higher-fidelity pair probabilistically, requiring two-way classical communication and reducing throughput.It also ties up buffer memory resources during the attempt.
- Network-level effects: For a specified fidelity threshold, throughput declines in stair steps with hop count while total work grows linearly between sudden increases at stair crossings.Nonpower-of-two hop counts also affect throughput and total work.
3 Problems and Solutions
The paper formulates quantum routing around a goal, a scalar link cost, and a path-cost function, then tests whether Dijkstra can rank heterogeneous repeater paths effectively. It emphasizes minimizing work while preserving throughput and compares several candidate link metrics.
- PS.1: Goal: The paper seeks agreement between detailed network simulations and an easy-to-calculate algorithmic cost for choosing effective paths.The intended setting is a heterogeneous quantum repeater network.
- PS.2–PS.3: Cost design: A routing algorithm requires a link-cost representation and a function that aggregates link costs into a path cost.The paper separates choosing the routing goal from defining link and path costs.
- PS.1: Goal: The routing objective is to minimize work along a path while secondarily selecting the highest-throughput single path at a target fidelity of F = 0.98.Work is evaluated using total measurement operations and total pulse count.
- PS.2: Link Cost: Candidate link costs include channel loss, inverse transmittance, pulse count, measurement count, and single-link Bell-pair generation time.BellGenT is measured in seconds per Bell pair and incorporates system factors.
- PS.2: Link Cost: BellGenT may provide more accurate estimates than pulse or measurement counts with little additional complexity because it incorporates system factors.The candidates differ in how much functionality they capture, from physical characteristics to reliable link behavior.
- PS.3: Path Cost Function: The central hypothesis is that Dijkstra’s algorithm can be used as-is when paired with an appropriate link cost, with path cost formed from the costs c_i of links in path P.The candidate naming convention includes Dijkstra/BellGenT.
4 Simulation and Results
Simulations of heterogeneous qubus repeater paths show that throughput is shaped mainly by purification and bottleneck links, while BellGenT-based Dijkstra generally tracks throughput and total work. The approach performs well overall but has technology and optimization limitations.
- Simulation setup: Four link qualities were simulated across forty-six paths of one to nine hops and all 256 four-hop combinations.The simulations model qubus physical entanglement and compare heterogeneous path configurations.
- Limitations: The simulations model only the qubus mechanism, whose basic form fails above about 5.5dB loss and roughly 30km over high-quality telecom fiber.The authors plan to test Dijkstra with other physical link types; path optimization also remains brute-force and can require delicate adjustments.
- Throughput behavior: Minor fidelity changes can trigger an additional purification round, generally reducing throughput by 50%.Throughput therefore changes discretely rather than smoothly as path conditions vary.
- Throughput behavior: Throughput is usually limited by the weakest link, although hop count, weak-link position, and path optimization can also affect performance.A single low-quality link makes other links nearly irrelevant in many paths, but exceptions occur.
- Path selection: 82.6% of unequal-cost pairs among the 256 four-hop paths were ordered by Dijkstra/BellGenT toward the higher-throughput path.For forty-six variable-length paths, the rate of correct choices was 81.6%; incorrect choices sometimes incurred penalties of 25% or more.
- Path costs: Path cost correlates with total work, with R^2 = 0.88 for both pulses and measurements on variable-length paths and R^2 = 0.81 and 0.99 on four-hop paths.The four-hop values correspond to pulses and measurements, respectively.
- Path costs: BellGenT uses seconds per Bell pair, while measurement count is selected to minimize total work along a path.Pulse and measurement counts correlate with work, but lower-level metrics are poor link-prioritization costs when throughput is the goal.
5 Conclusion
The paper evaluates Dijkstra-based path selection for heterogeneous quantum repeater networks and finds strong agreement between inverse-throughput path cost and simulated behavior, while identifying important scope boundaries.
- Real-world repeater networks require routing methods that handle physically heterogeneous links, complex topologies, and many possible paths.
- The study combines detailed simulations of quantum physics and classical messaging to evaluate Dijkstra-based path selection.
- Inverse throughput per hop produces strong correlation between calculated path cost and actual throughput, while total work also grows with non-bottleneck links.
- The simulations support using Dijkstra with reasonable computational effort for purify-and-swap quantum repeater networks.
- The results were obtained for uniform qubus technology with disparate operating conditions, not across different physical-layer repeater types.
- Whether the findings apply to error-corrected rather than purified repeater networks remains an open question.