Bayesian Optimization for Repeater Protocols

TL;DR

The paper tackles the problem of maximizing end-to-end secret-key rate in first-generation quantum repeater chains by extending the Li_2021 model to arbitrary, potentially heterogeneous node configurations. It leverages Bayesian optimization with a probabilistic encoding over protocol structures to efficiently search the vast protocol space, evaluating protocols via a time-resolved, Werner-state-based model of entanglement generation, swapping, and distillation. Key contributions include extending the simulation to heterogeneous chains, formalizing the protocol space as binary-tree structures with distillation budgets, and demonstrating that Bayesian optimization finds near-optimal protocols faster than brute-force, yielding practical design insights. Findings show that, in homogeneous regimes, the optimal protocols tend to be maximally symmetric and distillation is beneficial mainly for low-quality links, while in heterogeneous chains balancing waiting times across links is crucial; the framework provides scalable guidance for near-term quantum repeater deployments. The work highlights the practical impact of combining a rigorous probabilistic model with Bayesian search to optimize complex quantum network protocols under realistic hardware conditions.

Abstract

Efficiently distributing secret keys over long distances remains a critical challenge in the development of quantum networks. "First-generation" quantum repeater chains distribute entanglement by executing protocols composed of probabilistic entanglement generation, swapping and distillation operations. However, finding the protocol that maximizes the secret-key rate is difficult for two reasons. First, calculating the secretkey rate for a given protocol is non-trivial due to experimental imperfections and the probabilistic nature of the operations. Second, the protocol space rapidly grows with the number of nodes, and lacks any clear structure for efficient exploration. To address the first challenge, we build upon the efficient machinery developed by Li et al. [1] and we extend it, enabling numerical calculation of the secret-key rate for heterogeneous repeater chains with an arbitrary number of nodes. For navigating the large, unstructured space of repeater protocols, we implement a Bayesian optimization algorithm, which we find consistently returns the optimal result. Whenever comparisons are feasible, we validate its accuracy against results obtained through brute-force methods. Further, we use our framework to extract insight on how to maximize the efficiency of repeater protocols across varying node configurations and hardware conditions. Our results highlight the effectiveness of Bayesian optimization in exploring the potential of near-term quantum repeater chains.

Figures (8)

• Figure 1: Visual representation of a repeater protocol involving entanglement distillation and swapping. (a) All links are probabilistically generated. (b) Entanglement distillation is performed on the leftmost couple of nodes: in case of success, the two entangled links are distilled into a single one of higher quality. (c) Adjacent entangled links are probabilistically swapped, and then (d) the output link is swapped again to have end-to-end entanglement.
• Figure 2: Visual representation of a single repeater protocol, for a chain of $N=4$ nodes. Each vertex of the tree represents an entangled link in the chain. Leaves of the tree represent freshly generated links. In this case, we have $3$ leaves, which are all distilled once, as shown on their label. When two entangled links (vertices) are swapped, the output link is represented in the tree as their parent. The root represents the end-to-end link, on which two rounds of entanglement distillation are performed.
• Figure 3: Optimal protocol found by the Bayesian optimization algorithm for scenario C of Table \ref{['tab:hardware_regimes']}, with $N=5$ and $\beta=1$ (a) or $\beta=2$ (b).
• Figure 4: Optimal protocol found by the Bayesian optimization algorithm for scenario D of Table \ref{['tab:hardware_regimes']}, with $N=11$ and $\beta=2$.
• Figure 5: Performance of the optimal protocols found by the Bayesian optimization, in terms of the secret-key rate as a function of the initial Werner parameter $w_0$. For each value of $w_0$ we consider two settings, one allowing a maximum of $\beta=3$ rounds of distillation, and one allowing none. In (a) and (b) the optimal protocols are those in which distillation is performed only at link-level. In (a), entanglement generation of each link is followed by three rounds of distillation, while in (b) one round of distillation is performed. In (c) and (d), the optimal protocols do not perform entanglement distillation.