Operational-regime delineation for sequential entanglement generation with error detection (SEG-ED) quantum repeaters

The SEG-ED repeater architecture (Rey-Domínguez and Razavi, 2025) distributes entanglement hop by hop over a chain, encoding each link's qubits in three-qubit repetition codes and aborting a distribution round whenever a swapped syndrome detects an error, releasing all resources in a connectionless, statistical-multiplexing style. The source paper demonstrates continental-scale entanglement-based QKD feasibility at three discrete technology stages but leaves open the central design question it itself poses: the regimes of operation in which SEG-ED is beneficial are unclear — too-noisy equipment may make the abort frequency render it unusable, and the boundaries where unencoded sequential variants or parallel-generation variants dominate are unmapped. The problem is to delineate this operational regime: characterize secret-key-rate and resource-cost performance across the hardware/network parameter space, locate the transitions at which SEG-ED becomes impractical relative to SEG-noED, SEG-prob, and PEG-ED alternatives, and determine how these boundaries shift under stronger error-detection codes and realistic non-depolarizing noise. No resolution of this question is known in the subsequent literature.

Background

Long-distance fiber quantum communication is limited by exponential channel loss, which quantum repeaters overcome by dividing the path into shorter hops, creating entanglement on each hop, and joining the pieces by entanglement swapping (a joint Bell-state measurement that teleports entanglement across intermediate nodes). The classic design space runs from probabilistic repeaters (repeat-until-success link generation plus entanglement distillation), to encoded repeaters introduced by Jiang et al. (Phys. Rev. A 79, 032325 (2009)), in which each qubit is protected by a quantum error-correcting code as it traverses the chain. Jing, Alsina, and Razavi (Phys. Rev. Applied 14, 064037 (2020)) showed that encoded repeaters can use the code syndrome purely for error detection — discarding detected-bad rounds as an effective postselection — while still achieving useful quantum key distribution (QKD) rates, avoiding full error correction. Rey-Domínguez and Razavi (arXiv:2508.16310) proposed SEG-ED, combining two ideas: sequential entanglement generation (SEG), in which entanglement is generated and swapped hop by hop as resources become free rather than reserving the whole path (contrast with parallel entanglement generation, PEG, which attempts all links simultaneously along a reserved path), and error detection (ED) via the three-qubit repetition code (3QRC — the smallest code, three physical qubits per logical qubit, that flags bit-flip errors; used here only to detect, not correct). In SEG-ED, each hop generates an encoded Bell pair (link-level entanglement generation, LLEG, succeeds with probability $p_{\mathrm{gen}}=p_{\mathrm{cou}}^{2}\eta_{D}^{2}10^{-\alpha_{\mathrm{ch}}L_0/10}/2$ , where $p_{\mathrm{cou}}$ is the photon coupling probability, $\eta_D$ the detector efficiency, $\alpha_{\mathrm{ch}}$ the fiber attenuation, and $L_0$ the hop length), then a logical swap extends the entanglement; if a syndrome signals an error, the round aborts immediately and every node's memory is released for other users — the quantum analogue of store-and-forward packet switching, motivated by fairness and interoperability with classical telecom infrastructure. Performance is benchmarked by the asymptotic secret key rate $K$ of BBM92 QKD, the entanglement-based variant of the BB84 protocol in which Alice and Bob each measure their half of a distributed entangled pair in randomly chosen bases (its secret fraction is $r_\infty=\max\{0,1-h_2(e_Z)-h_2(e_X)\}$ for phase- and bit-error rates $e_X,e_Z$ and binary entropy $h_2$ ) and by a cost metric $C_K=\frac{4N_{\mathrm{mux}}}{K}\left[2+\sum_{i=2}^{N_r}\prod_{j=1}^{i-1}p_{\mathrm{LSWP},j}\right]$ — average quantum memories occupied per unit key rate, with $N_{\mathrm{mux}}$ the number of multiplexed memories per node, $N_r$ repeater nodes, and $p_{\mathrm{LSWP},j}$ the probability the logical swap at node $j$ passes error detection. The relevant hardware parameters are the two-qubit gate error rate at repeater nodes, the single-qubit measurement error probability $\delta$ , the fidelity $F_0$ of the Werner state produced by a successful LLEG round, the memory coherence time $T_{\mathrm{coh}}$ (memory decoherence is modeled as depolarization with probability $1-e^{-\tau/T_{\mathrm{coh}}}$ after storage time $\tau$ ), $p_{\mathrm{cou}}$ , $\eta_D$ , $N_{\mathrm{mux}}$ , $L_0$ , and the total distance. The paper compares SEG-ED against unencoded sequential variants (SEG-noED with deterministic noisy swaps; SEG-prob with probabilistic heralded swaps) and a PEG variant with error detection (PEG-ED), at three discrete technology stages, finding for instance that at the most optimistic stage SEG-noED outperforms both SEG-ED and PEG-ED up to roughly 5000 km — hinting that error detection may be unnecessary once gate errors fall far enough. What is missing is the map itself: no continuous or systematic delineation of where abort frequency kills the scheme, where each alternative dominates, and how these boundaries move under stronger detection codes and non-depolarizing (hardware-specific) noise, which the paper's depolarizing model does not capture.

Problem Statement

For a repeater chain with hop length $L_0$ and total distance $L$ , delineate the operational regime of SEG-ED — sequential hop-by-hop logical entanglement generation with 3QRC encoding and abort-upon-error-detection — by characterizing the achievable entanglement-based QKD secret key rate $K$ and the resource-occupation cost $C_K$ as functions of the hardware and network parameters (two-qubit gate error rate at repeater nodes, single-qubit measurement error probability $\delta$ , initial LLEG Werner-state fidelity $F_0$ , memory coherence time $T_{\mathrm{coh}}$ , coupling probability $p_{\mathrm{cou}}$ , detector efficiency $\eta_D$ , multiplexing factor $N_{\mathrm{mux}}$ , channel attenuation $\alpha_{\mathrm{ch}}$ , $L_0$ , and $L$ ). Throughout, a scheme is said to dominate another when it is no worse on both metrics and strictly better on at least one — i.e. Pareto dominance over the pair $(K, C_K)$ ; SEG-ED is called beneficial in the parameter region where no alternative dominates it, and this rule is applied uniformly to all four schemes. Specifically: (i) locate the boundaries in this parameter space at which the frequency of error-detection aborts makes SEG-ED impractical — for instance, where the unencoded sequential variants SEG-noED and SEG-prob dominate it (their key rate exceeds that of SEG-ED while their resource-occupation cost $C_K$ is lower); (ii) locate the transitions at which parallel-generation-based operation (PEG-type schemes) becomes preferable, refining the paper's heuristic estimate $(3/2)^{\hat n}\tau_{\mathrm{hop}}$ for the round duration of advance-generation (pre-generated) parallel schemes — with $\hat n$ the average number of node pairs not yet sharing entanglement at the start of a round and $\tau_{\mathrm{hop}}$ the average waiting time to generate one encoded Bell pair at a hop — into a quantified comparison; and (iii) determine how these boundaries shift when stronger error-detection codes replace 3QRC (e.g., larger repetition codes or stabilizer codes) and when realistic non-depolarizing noise models (e.g., biased/phase-dominant or correlated error processes) replace the depolarizing model. The deliverable is a quantitative regime map — analytic boundary expressions and/or pinned simulation-based phase boundaries — that a quantum-network designer can use to decide, for given hardware, whether SEG-ED is the right architecture choice. All three of (i)–(iii) are required, and the boundaries must be characterized as functions of the parameters listed above rather than at isolated operating points; a representative set of parameter slices that resolves (i)–(iii) across that list suffices, and exhaustive sampling is not required.

References

J. Rey-Domínguez and M. Razavi, "Rethinking Quantum Repeaters: Balancing Scalability, Feasibility, and Interoperability," arXiv:2508.16310 (2025). DOI: 10.48550/arXiv.2508.16310, https://arxiv.org/abs/2508.16310
K. Campbell, A. Lawey, and M. Razavi, "Combatting noise in near-term quantum data centres," Quantum Science and Technology 11, 045011 (2026). DOI: 10.1088/2058-9565/ae978f, arXiv:2601.14845, https://arxiv.org/abs/2601.14845
L. Jiang, J. M. Taylor, K. Nemoto, W. J. Munro, R. Van Meter, and M. D. Lukin, "Quantum repeater with encoding," Physical Review A 79, 032325 (2009). DOI: 10.1103/PhysRevA.79.032325, https://doi.org/10.1103/PhysRevA.79.032325
Y. Jing, D. Alsina, and M. Razavi, "Quantum key distribution over quantum repeaters with encoding: Using error detection as an effective postselection tool," Physical Review Applied 14, 064037 (2020). DOI: 10.1103/PhysRevApplied.14.064037, https://doi.org/10.1103/PhysRevApplied.14.064037