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Entanglement Swapping Scheduling for Quantum Repeater Chains under Decoherence during Classical Communications

This paper presents an analytical model demonstrating that decoherence during classical heralding creates a trade-off between increased entanglement generation and memory degradation, revealing an optimal number of repeaters for maximizing long-distance entanglement rates in first-generation quantum networks.

Original authors: Jorge Benito Díaz, Andrés Agustí, Jabier Faba, Laura Ortiz, Luis Robledo, Vicente Martín

Published 2026-10-07

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CC BY 4.0

Original authors: Jorge Benito Díaz, Andrés Agustí, Jabier Faba, Laura Ortiz, Luis Robledo, Vicente Martín

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The dream of a quantum internet relies on a peculiar resource: entanglement. This is a connection between particles that allows them to share information instantly, regardless of the distance separating them. In theory, this could enable unhackable communication and powerful new computers. However, there is a major obstacle. When scientists try to send these particles through optical fibers, the signal fades away quickly, much like a whisper dying out in a long hallway. To overcome this, engineers plan to use "repeaters," devices placed along the path that catch the fading signal, store it, and pass it on. But these devices are not perfect. They must hold the delicate quantum information in their memory while waiting for confirmation that the previous step worked. During this waiting time, the information naturally degrades, or "decoheres," like a sandcastle slowly losing its shape in the wind. The central question for building this future network is how many of these repeaters to use and how to coordinate them to get the best results before the information disappears.

A team of researchers has developed a new way to calculate the best balance for these networks. They created a mathematical model that simulates how a chain of repeaters behaves when classical signals—ordinary messages sent to confirm success—take time to travel between the devices. The study focuses on the tension between two opposing forces. On one hand, adding more repeaters shortens the distance each signal must travel, making it easier to generate the initial connections. On the other hand, every new repeater adds more memory units that must hold the information while waiting for the entire chain to be ready. The longer the information sits in memory, the more it degrades. The researchers found that there is a sweet spot: a specific, optimal number of repeaters for any given distance. Adding more devices beyond this point actually hurts performance because the accumulated waiting time and memory decay outweigh the benefits of shorter distances.

The team tested three different ways to organize the swapping process, which is the act of linking short segments together to form a long connection. The first method, called parallel scheduling, attempts to link all segments at the exact same time. The second, binary-tree scheduling, builds the connection in steps, linking pairs of segments, then linking those results together, and so on. The third, hybrid scheduling, mixes these two approaches. The study revealed that the best strategy depends entirely on the distance and the quality of the memory. For shorter distances, such as within a city, the simplest approach works best. In these cases, a small number of repeaters, perhaps just one or two, provides the highest rate of successful connections, and doing everything at once is the most efficient method.

However, for much longer distances, such as between cities, the rules change. The researchers simulated a scenario spanning five hundred kilometers and found that the optimal number of repeaters jumps significantly, settling around ten or eleven. In these long-distance scenarios, the most efficient method is not to do everything at once, nor to build it strictly step-by-step, but to use a hybrid approach. This method groups the repeaters into blocks, processes them, and then links the blocks together. This hierarchical organization helps manage the waiting times and preserves the quality of the connection better than the other methods. The study suggests that for very long networks, a rigid, single strategy will not work; instead, the network architecture must be flexible enough to support different scheduling methods depending on the specific distance and the capabilities of the memory devices.

The findings also highlight that the speed of the classical communication system is just as critical as the quantum hardware itself. Because the repeaters must wait for a confirmation signal before proceeding, the time it takes for that signal to travel determines how long the quantum information must sit in memory. The researchers showed that this delay is a major factor in how much the information degrades. Consequently, the design of future quantum networks cannot focus solely on the quantum components; it must also account for the timing of the control signals. The study concludes that there is no single "best" number of repeaters or a single best way to schedule them for every situation. Instead, network designers must treat the number of repeaters and the scheduling method as variables to be tuned carefully, balancing the need for short links against the risk of memory decay, to ensure the quantum internet can function effectively over real-world distances.

Technical Summary: Entanglement Swapping Scheduling for Quantum Repeater Chains under Decoherence during Classical Communications

Problem Statement
First-generation quantum repeaters are critical for distributing entanglement over long distances by mitigating optical fiber attenuation. These systems rely on two probabilistic operations: Entanglement Generation (EG) between adjacent nodes and Entanglement Swapping (ES) to extend entanglement across the chain. A significant challenge in designing these networks is the interplay between the benefits of adding more repeaters (shorter links, higher EG probability) and the detrimental effects of quantum memory decoherence.

Crucially, both EG and ES are heralded operations requiring classical communication to confirm success. This introduces latency during which stored entangled pairs undergo decoherence. The paper addresses the lack of analytical models that rigorously account for this "exposure" time—the cumulative duration entangled states remain in memory while waiting for classical confirmation signals. Furthermore, it investigates how different scheduling strategies for ES operations (when to perform swaps) interact with memory decoherence and classical signaling delays to determine the overall network performance.

Methodology
The authors develop a hardware-agnostic analytical framework to model a homogeneous linear chain of $N$ quantum repeaters connecting two end nodes over a total distance $L_{tot}$ .

Physical Model:
- Entanglement Generation: Modeled as a geometric process where each elementary link attempts EG until success. Success probability depends on link length and photon loss.
- Memory Decoherence: Quantum memories are characterized by a coherence time $\tau$ . Successful pairs are stored as Werner states. The Werner parameter decays exponentially with time spent in memory.
- Heralding Delay: The model explicitly incorporates the time required for classical signals to propagate. This delay contributes to the "memory exposure" ( $\Theta$ ), defined as the cumulative sum of storage times for all states contributing to the final end-to-end (E2E) pair.
Scheduling Strategies:
The study focuses on "prepare-and-swap" policies, where all elementary links must be generated before swapping begins. Three strategies are compared:
- Binary-Tree: Hierarchical composition of links via successive stages of 1-ES operations.
- Parallel: All $N$ repeaters perform ES simultaneously in a single stage ( $N$ -ES).
- Hybrid: The chain is divided into blocks; internal blocks perform parallel swaps, followed by a final stage to connect the blocks.
Performance Metric:
The authors utilize Entanglement Rate ( $R$ ), defined as the expected distributed entanglement per unit time.
- Entanglement Quantity: Calculated using Logarithmic Negativity, which sums the quality of entanglement over all generated pairs.
- Time Accounting: Includes the duration of the generation phase, all swapping stages, and the classical signaling overhead required for readiness and failure detection.
- Early-Abort Policy: If an ES operation fails at an intermediate stage, the protocol aborts immediately, saving time that would otherwise be spent on subsequent failed stages.

Key Contributions

Analytical Model of Exposure: The paper introduces the concept of "memory exposure" ( $\Theta$ ) to quantify the cumulative decoherence suffered by entangled pairs due to both asynchronous generation waiting times and the specific organization of the swapping phase.
Optimal Repeater Count: The model demonstrates that there is a non-trivial optimal number of repeaters ( $N$ ) for a given distance and memory coherence time. Increasing $N$ improves link generation rates but eventually degrades performance due to the additive impact of memory decoherence and the increased probability of swapping failures.
Strategy Comparison: The work provides a comparative analysis of Binary-Tree, Parallel, and Hybrid scheduling strategies, showing that the optimal strategy depends on the specific architectural regime (distance and memory quality).

Results
The study evaluates the model across two scenarios: a metropolitan scale ( $L_{tot} = 60$ km) and an intermetropolitan scale ( $L_{tot} = 500$ km).

Metropolitan Scenario: For shorter distances, the optimal number of repeaters is small (typically $N=1$ or $N=2$ ). In this regime, Parallel scheduling generally yields the highest entanglement rate around the optimum. The benefit of adding more repeaters diminishes rapidly as memory decoherence increases.
Intermetropolitan Scenario: For longer distances, a clear interior maximum exists for the entanglement rate as a function of $N$ $N$ . The optimal $N$ $N$ shifts based on memory coherence time (e.g., $N \approx 10$ $N \approx 10$ –$11$ for $\tau=100$ $τ = 100$ ms, shifting to lower $N$ $N$ for longer $\tau$ $τ$ ).
- In this regime, Hybrid scheduling often outperforms Parallel and Binary-Tree strategies, particularly around the optimal repeater count for longer memory regimes.
- Parallel scheduling becomes increasingly penalized as $N$ grows due to the high communication span required for a single-stage swap.
- Binary-Tree scheduling approaches Hybrid performance as memory coherence times increase.

Significance and Claims
The paper claims that its findings offer critical guidance for the design of future quantum internet architectures, specifically regarding:

Repeater Density: Repeater density should be treated as a design variable to be optimized rather than maximized. Denser deployment is not inherently advantageous under finite-memory conditions.
Scheduling Flexibility: No single ES scheduling strategy is optimal across all regimes. Future architectures should support the implementation of multiple strategies (e.g., switching between parallel and hybrid approaches) rather than relying on a fixed protocol.
Classical-Quantum Coupling: Classical signaling delays are an integral part of the network design, directly coupling the control plane to memory decoherence. The "exposure" model highlights that the timing of classical confirmation is as critical as the quantum operations themselves.

The authors conclude that their framework allows for the extraction of novel conclusions regarding the trade-offs between inter-repeater distance, swapping timings, and memory coherence times, providing a foundation for optimizing quantum network performance in the presence of realistic decoherence and latency.

Technical Summary: Entanglement Swapping Scheduling for Quantum Repeater Chains under Decoherence during Classical Communications

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