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IEEE International Conference on Quantum Computing and Engineering (QCE)

Towards Optimal Orders for Entanglement Swapping in Path Graphs: A Greedy Approach

Van Sy Mai, Abderrahim Amlou, Amar Abane, Abdella Battou

Published Aug 30, 2025• 3 citations
Publisher

Abstract

This paper considers the problem of finding an optimal order for entanglement swapping in a heterogeneous path of quantum repeaters so as to maximize the path throughput defined as the delivery rate of end-to-end entanglements. The primary difficulty in addressing this problem lies in the vast array of possible swapping orders for large paths and the complexity of the expected throughput, which depends on the attributes of each node and edge along the path, as well as the order of swapping. To cope with these issues, we first propose simple approximations in estimating the swapping outcome between two entanglement distributions that can run in constant time, thereby providing an efficient approach for evaluating and comparing different swapping orders, allowing us to solve the problem exactly for small paths. Second, as the number of possible orders grows exponentially with the number of repeaters in the path, we develop an efficient heuristic based on the greedy selection of nodes to sequentially perform swaps according to their swapping scores, defined as the expected number of entanglements resulting from their swaps. The scores are local but dynamic in the sense that they depend not just on the entanglement distributions available on the path but also on prior swapping decisions. Finally, we illustrate the efficiency and effectiveness of our proposed model and approach through extensive experimentation conducted using a general quantum network simulator.

DOI
10.1109/qce65121.2025.00138

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This paper considers the problem of finding an optimal order for entanglement swapping in a heterogeneous path of quantum repeaters so as to maximize the path throughput defined as the delivery rate of end-to-end entanglements. The primary difficulty in addressing this problem lies in the vast array of possible swapping orders for large paths and the complexity of the expected throughput, which depends on the attributes of each node and edge along the path, as well as the order of swapping. To cope with these issues, we first propose simple approximations in estimating the swapping outcome between two entanglement distributions that can run in constant time, thereby providing an efficient approach for evaluating and comparing different swapping orders, allowing us to solve the problem exactly for small paths. Second, as the number of possible orders grows exponentially with the number of repeaters in the path, we develop an efficient heuristic based on the greedy selection of nodes to sequentially perform swaps according to their swapping scores, defined as the expected number of entanglements resulting from their swaps. The scores are local but dynamic in the sense that they depend not just on the entanglement distributions available on the path but also on prior swapping decisions. Finally, we illustrate the efficiency and effectiveness of our proposed model and approach through extensive experimentation conducted using a general quantum network simulator.
Entanglement distribution via probabilistic entanglement swapping across a quantum repeater chain connecting two quantum nodes is a challenging problem. The difficulty lies in the exponential number of possible swapping structures within the repeater chain, necessitating efficient search algorithms, especially as the chain length increases. In this paper, we first explore the algorithmic design to facilitate the search for the optimal swapping structure along a repeater chain, aiming to maximize the entanglement distribution rate. Second, we examine the computational complexities of various algorithms and find that prior approaches exhibit excessively high complexities. Thus, we propose an efficient dynamic programming-based algorithm, FastHED, that leverages heuristics to expedite the search for the optimal swapping structure. Our theoretical analysis reveals that the upper bound of the proposed algorithm’s computational complexity isO(n(log⁡n)3)O(n(\log n)^{3})O(n(logn)3)(more precisely,O(n(log⁡n)2log⁡log⁡n)O(n(\log n)^{2} \log \log n)O(n(logn)2loglogn)whenn≤229n\le 2^{29}n≤229), a significant improvement over the existing algorithm with a complexity ofO(n2log⁡n)O(n^{2} \log n)O(n2logn), where n denotes the repeater chain’s length. Additionally, we design a best-first framework to evaluate the performance of different algorithms. Numerical results show that our algorithm achieves a higher average entanglement distribution rate than existing algorithms.
Remote entanglement distribution plays a crucial role in large-scale quantum networks, and the key enabler for entanglement distribution is quantum routers (or repeaters) that can extend the entanglement transmission distance. However, the performance of quantum routers is far from perfect yet. Amongst the causes, the limited quantum memories in quantum routers largely affect the rate and efficiency of entanglement distribution. To overcome this challenge, this paper presents a new modeling for the maximization of entanglement distribution rate (EDR) on a memory-limited path, which is then transformed into entanglement generation and swapping sub-problems. We propose a greedy algorithm for short-distance entanglement generation so that the quantum memories can be efficiently used. As for the entanglement swapping sub-problem, we model it using an Entanglement Graph (EG), whose solution is yet found to be at least NP-complete. In light of it, we propose a heuristic algorithm by dividing the original EG into several sub-problems, each of which can be solved using dynamic programming (DP) in polynomial time. By conducting simulations, the results show that our proposed scheme can achieve a high EDR, and the developed algorithm has a polynomial-time upper bound and reasonable average runtime complexity.
Due to the fact that quantum networks can provide higher security in communications, many related issues have been presented and addressed recently. One of the most important issues is to study the routing in quantum networks with multiple source-destination pairs while maximizing the total end-to-end entanglement rate. In previous studies, the shortest paths are typically selected as the entanglement paths for a source-destination pair to increase the total end-to-end entanglement rate. However, this approach is only efficient once the network resources are sufficient. Moreover, source-destination pairs only require a specific entanglement rate rather than the shortest paths. Therefore, in this paper, we suppose that each source-destination pair requires an upper bound of the entanglement rate. Under this assumption, a novel problem is proposed to construct entanglement paths for a set of multiple source-destination pairs such that the total end-to-end entanglement rate is maximized. We then propose a greedy algorithm to solve the problem. Finally, the simulation is conducted to evaluate the performance of the proposed algorithm.
Quantum repeater networks distribute entanglement over lossy links while many users share a limited pool of entangled pairs. Most existing routing schemes either always use a single best path or rely on global optimizations that are hard to run in real time. Here we propose and analyze a simple alternative: a stochastic multipath rule in which each entanglement request is sent at random along one of several edge-disjoint repeater paths, with a single parameter that controls the bias between shorter and longer routes. Using a distance-dependent lossy network model with finite per-link capacities and probabilistic entanglement swapping, we develop an analytic description of the resulting end-to-end entanglement rate as a function of this bias and validate it with large-scale numerical simulations. We find that an intermediate bias consistently outperforms both deterministic extremes across distances, traffic patterns, attenuation, swapping noise, and congestion, bringing the rate close to simple capacity upper bounds and making link usage more even across networks. These results identify stochastic multipath routing as a lightweight classical control strategy for boosting performance and scalability in near-term quantum repeater networks.
In this paper, we study the properties of path metrics of an entanglement path for a given entanglement swapping order of the path. We show how to efficiently compute the path metrics of an entanglement path for any given swapping order. We show that different entanglement swapping orders for the same path can lead to different expected throughputs. A key finding is that the binary operator corresponding to entanglement swapping along a path is not associative. We further show that the problem of computing an s-t path with maximum expected throughput under any entanglement swapping order does not have the subpath optimality property, which is a key property most path finding algorithms such as Dijkstra’s algorithm rely on. We use extensive simulations to validate our theoretical findings.
The fundamental technology for the construction of large-scale quantum networks lies in the efficient distribution of remote entanglement among quantum nodes. Entanglement routing tackles the challenge of path selection between two distant end nodes and realizes entanglement distribution through multi-hop entanglement swapping. Given the inherent probability of swapping, entanglement distribution flow always presents inevitable losses. The existing routing works adopt a parallel swapping mode, which only considers the final end-to-end distribution loss effect. This lag of failure detection and complete independence between swapping reduce distribution efficiency. We observe that the sequential swapping mode can improve this defect, but it is not thorough. Thus, in this paper, we explore the entanglement flow loss effect in the sequential entanglement swapping (Loss-SES) process in order to perceive failures earlier and save network resources. Subsequently, we introduce a novel metric to quantify path performance, namely the expected entanglement throughput (EET). Building upon EET, we propose the entanglement routing algorithm MaxEET adapted to the Loss-SES process to maximize EET. For efficient deployment, in the single-request scenario, we determine the optimal routing initiation direction by introducing a bidirectional routing mechanism. In the multi-request scenario, we design a greedy and contention-free iterative routing algorithm to exhaust resources. Based on extensive simulations, the results show the superiority of our proposed Loss-SES mode compared to other modes in terms of throughput, fairness, and utilization.