We study the problem of entanglement distribution in terms of maximizing a utility function that captures the total fidelity of end-to-end entanglements in a two-link linear quantum network with a source, a repeater, and a destination. The nodes have several quantum memories, and the problem is how to coordinate entanglement purification in each of the links, and entanglement swapping across links so as to achieve the goal above. We show that entanglement swapping (i.e, deciding on the pair of qubits from each link to perform swapping on) is equivalent to finding a max-weight matching on a bipartite graph. Further, entanglement purification (i.e, deciding which pairs of qubits in a link will undergo purification) is equivalent to finding a max-weight matching on a non-bipartite graph. We propose two polynomial algorithms, the Purify-then-Swap (PtS) and the Swap-then-Purify (StP) ones, where the decisions about purification and swapping are taken with different order. Numerical results show that PtS performs better than StP, and that the omission of purification in StP gives substantial benefits.
We study the problem of entanglement distribution in terms of maximizing a utility function that captures the total fidelity of end-to-end entanglements in a two-link linear quantum network with a source, a repeater, and a destination. The nodes have several quantum memories, and the problem is how to coordinate entanglement purification in each of the links, and entanglement swapping across links so as to achieve the goal above. We show that entanglement swapping (i.e, deciding on the pair of qubits from each link to perform swapping on) is equivalent to finding a max-weight matching on a bipartite graph. Further, entanglement purification (i.e, deciding which pairs of qubits in a link will undergo purification) is equivalent to finding a max-weight matching on a non-bipartite graph. We propose two polynomial algorithms, the Purify-then-Swap (PtS) and the Swap-then-Purify (StP) ones, where the decisions about purification and swapping are taken with different order. Numerical results show that PtS performs better than StP, and that the omission of purification in StP gives substantial benefits.
Abstract We address the problem of efficiently distributing high-fidelity entangled states across the nodes of a quantum network—a fundamental requirement for quantum communication, distributed quantum computing, and emerging quantum internet applications. We consider a repeater-based architecture that combines entanglement swapping (or fusion) with purification operations, which enhance fidelity by consuming additional entangled pairs. While prior work has explored routing and limited purification strategies, we present the first algorithms for fidelity-constrained entanglement distribution that are near-optimal and compatible with arbitrary purification models. Our approach introduces the notion of purification-augmented entanglement trees, and we develop dynamic programming (DP) algorithms to construct such trees that maximize the generation rate of entangled pairs (EPs) under fidelity constraints. We further extend our framework to a linear programming (LP)–based flow model that enables simultaneous generation of EPs across multiple source-destination pairs. For multipartite entanglement, we generalize our methods to GHZ and arbitrary graph states, supported by novel analytical models for fidelity evolution under fusion and purification. We evaluate our techniques using the NetSquid quantum network simulator and observe consistent performance improvements over prior approaches—achieving an average 50\% increase in generation rate, and up to 100-150\% in some scenarios.
Entanglement routing establishes entangled pairs between distant nodes of a quantum network by purifying and swapping pairs generated on elementary links. Existing methods typically restrict the decision space along two axes: the operation order, often fixed to purify-then-swap (PtS), and the purification schedule of each link, often restricted to pumping. Focusing on linear repeater chains with finite link capacities, we relax both restrictions and study the maximization of the expected end-to-end throughput subject to a fidelity threshold under two quantum-noise models. We develop a unified capacity-aware optimization framework comprising an exact mixed-integer linear program (MILP) under PtS, instantiated with either pumping or general tree purification schedules, and an exact dynamic program (DP) over arbitrary operation orders. Under symmetric Pauli noise, we prove that pumping converges to a fidelity strictly below unity, imposing a capacity-independent feasibility bound and a maximum chain length on every pumping-based PtS method, whereas general tree schedules yield a capacity-dependent bound. We further prove that post-swap purification never increases the throughput, so a free operation order can only increase feasibility. Numerical evaluations confirm the bounds: tree schedules serve over 90\% of the requests that pumping cannot at moderate fidelity thresholds; within the pumping class, a free operation order recovers much of this advantage; but once links use tree schedules, the tree-based MILP and the order-exact DP serve identical request sets throughout. Operation order and purification schedule are thus substitutes, with the purification schedule the dominant factor governing feasibility under symmetric Pauli noise.
Entanglement purification with two entangled resource pairs is widely employed in the literature on quantum repeater networks to counteract fidelity degradation introduced by noisy quantum memories and entanglement swapping across multiple hops. Standard purification protocols assume both resource pairs carry identical fidelity. In practice, entanglement generation is stochastic, the two resource pairs are heralded at different times, and so the first pair decoheres in memory while the second is being generated. Thus a fidelity asymmetry is a structural feature of any network operating under realistic memory conditions, leading to the question: when is it beneficial to perform purification? We derive a closed-form fidelity asymmetry tolerance delta(F) that governs whether a purification attempt is beneficial. We determine a universal upper bound delta_max of approximately 0.076 beyond which purification is always counterproductive. Our simulations show that with exponential memory decoherence, purification yields benefits in only approximately 14% of purification attempts on two resource pairs in a two-hop repeater chain. We define three network objectives: fidelity only, time only, and a combination of time and fidelity, to deliver end-to-end entanglement. We show that when the application fidelity requirement is achievable through swapping alone, no-purification is the superior policy, with its advantage increasing with the number of hops. When the fidelity requirement cannot be met with swapping alone and purification is necessary, to be effective, it must be conditioned on delta(F) between resource pairs. We introduce DeltaPurify, a policy that conditions purification decisions on local fidelity information, and show it reduces time-to-serve relative to both naive purification and no-purification across several fidelity thresholds and hops of a repeater chain.
Swapping and purification are the two fundamental building blocks for high-fidelity entanglement distribution in multi-hop quantum networks. Unfortunately, it is still a mystery how they intertwine with each other to affect the fidelity and cost of end-to-end entanglements. Current scheduling algorithms consider this problem under relatively limited assumptions and a critical yet unjustified conjecture. In this work, we first consider more general assumptions with operation failures and, accordingly, extend a tree-based modeling for joint swapping and purification. Then, we analytically prove the previous conjecture that the optimal strategy underBinary systemis always to purify the entanglements before any swapping. This sheds light on the protocol and device design for quantum networks. We then further propose a tree-based algorithm, which can efficiently schedule swapping and purification along a path for bothBinaryandWerner systems. Extensive simulations of the proposed method against state-of-the-art solutions show that our method uses fewer entanglements to establish qualified end-to-end entanglements, and thus achieves higher network throughput.
Entanglement-based quantum networks provide unconditionally secure quantum communication by distributing end-to-end entangled pairs. To reach such goal, entanglement swapping and purification are two necessary operations, which can expand the entanglement distance and improve the fidelity of entangled pairs, respectively. During the end-to-end entanglement distribution process, the execution sequence of these two operations has a significant impact on the final performance, such as throughput and fidelity. Thus, how to schedule the sequence of swapping and purification becomes a critical problem in quantum networks. Our work focuses on such scheduling problem and we devise an efficient solution with fidelity maintenance. In our article, we first propose a concurrent swapping method for any-length path to reduce delay. Secondly, by modeling the scheduling sequence as a permutation problem, we obtain an optimal scheduling principle called “prioritized purification (PP)”. Finally, to meet the fidelity requirement of the end-to-end entangled pairs, we propose a fidelity maintenance scheme based on PP (FMSPP). The simulation results show that the proposed FMSPP can achieve higher distribution rates and less resource overhead than other scheduling solutions, and the advantages are more significant when there are more hops on the path.
End-to-end entanglement distribution in multi-hop quantum networks experiences fidelity degradation as entanglement swapping progresses over noisy and heterogeneous channels. In this paper, we propose a virtual-link purification strategy in which multiple EPR pairs are used at each stage to create virtual links between non-adjacent nodes and then purified in cascade using symmetric purification. The approach allows parameterizing the number of links per stage and follows a hierarchical design, with a more complete first stage and simplified purification in subsequent stages. We implement a time-slot-based orchestrator and compare the proposal against conventional swapping under Werner and bit-flip noise models. The results indicate higher end-to-end fidelity with virtual-link purification.