Routing in Non-Isotonic Quantum Networks

TL;DR

This work tackles pathfinding in quantum-repeater networks where the end-to-end utility, notably the secret-key rate $SKR_{F,T}$, is non-isotonic and undermines traditional routing like Dijkstra. It introduces two best-first-search algorithms (BeFS-EXACT and BeFS-HEURISTIC) that use destination-aware merit functions to reliably and quickly locate high-quality paths, with BeFS-EXACT guaranteed to be optimal for swap-ASAP repeaters. To accommodate broader models and tradeoffs, the authors also present two metaheuristics, simulated annealing and genetic algorithms, enabling tunable balance between solution quality and computational overhead. Extensive experiments on Waxman graphs show BeFS-HEURISTIC often yields near-optimal SKR with sublinear query growth, while metaheuristics provide robust, model-agnostic performance. The results demonstrate a practical, scalable framework for routing in realistic quantum networks and highlight the generality of non-isotonic routing challenges across repeater models.

Abstract

Optimal routing in quantum-repeater networks requires finding the best path that connects a pair of end nodes. Most previous work on routing in quantum networks assumes utility functions that are isotonic, meaning that the ordering of two paths does not change when extending both with the same edge. However, we show that utility functions that take into account both the rate and quality of the entanglement generation (e.g., the secret-key rate) are often non-isotonic. This makes pathfinding difficult as classical algorithms such as Dijkstra's become unsuitable, with the state of the art for quantum networks being an exhaustive search over all possible paths. In this work we present improved algorithms. First, we present two best-first-search algorithms that use destination-aware merit functions for faster convergence. One of these provably finds the best path, while the other uses heuristics to achieve an effectively sublinear scaling of the query count in the network size while in practice always finding a close-to-optimal path. Second, we present metaheuristic algorithms (simulated annealing and a genetic algorithm) that enable tuning a tradeoff between path quality and computational overhead. While we focus on swap-ASAP quantum repeaters for concreteness, our algorithms are readily generalized to different repeater schemes and models.

Figures (14)

• Figure 1: An example of a randomly generated quantum network (procedure explained in Sec. \ref{['sec:problem_statement']}), where blue squares indicate the end nodes that want to share entanglement and the orange circles represent quantum repeaters. The solid thick black line is the best path through the network, realizing a secret-key rate for quantum-key distribution (see Sec. \ref{['sec:problem_statement']}) of 6.4 Hz, found using our BeFS-EXACT algorithm (see Sec. \ref{['sec:deterministic_algs']}). On the other hand, the dashed thick black line is the path found using an algorithm (the extended Dijkstra's algorithm shi2020a) that assumes an isotonic utility function, which the secret-key rate is not in our repeater model. Over this path, a secret key can only be created at 0.17 Hz. The model parameters (explained in Sec. \ref{['sec:problem_statement']}) are a link fidelity of $F = 0.96$ and coherence time of $T = 10$ s.
• Figure 2: Example of non-isotonic behavior in quantum-repeater networks. The utility function, the secret-key rate $\text{SKR}_{F, T}$ for link fidelity $F=0.94$ and coherence time $T=10$ s, is approximately 32 Hz for path $p_1 = (\text{A}, \text{C})$ and 59 Hz for path $p_2 = (\text{A}, \text{B}, \text{C})$, making $p_2$ the better path between A and C. However, when we append edge $e = (\text{C}, \text{D})$ to both paths, we find approximately 13 Hz for path $p_1 \oplus e$ but exactly 0 for path $p_2 \oplus e$, making $p_1 \oplus e$ the better path between nodes A and D. The reason why path $p_2 \oplus e$ has zero utility is because with three links that each have a fidelity as low as $F=0.94$, the error rates in the end-to-end entangled states exceed the tolerance of the BBM92 protocol.
• Figure 3: Plot of the relative SKR inefficiency (defined as $\left.\left(\text{SKR}_\text{best} - \text{SKR}_\text{dijk}\right)\right/\left(\text{SKR}_{\text{best}}\right)$), where $\text{SKR}_\text{best}$ is the SKR of the best path and $\text{SKR}_\text{dijk}$ is the SKR of the path found by the extended Dijkstra's algorithm. We plot the relative SKR inefficiency for several link fidelities $F$ and coherence times $T$, where each point was sampled over 100 random Waxman graphs with $\alpha = 0.5$, $\beta = 0.9$, and $L=300\mathrm{km}$. Dijkstra's algorithm always finds the optimal path if $\text{SKR}_{F, T}$ is isotonic, and hence any deviation is evidence of non-isotonicity.
• Figure 4: The path tree for a fully-connected graph with 5 nodes starting at the bottom left blue node. The circles are non-leaf nodes, and the squares are leaf nodes. The path tree of a pathfinding problem is a tree structure imposed on the set of all paths starting at the start node, and in this diagram, each path is depicted inside of a node in the tree. Notice that parent paths and child paths differ by a single edge.
• Figure 5: Examples of the two types of path mutations used in the simulated annealing and genetic algorithms.

Theorems & Definitions (28)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof