Stochastic Multipath Routing for High-Throughput Entanglement Distribution in Quantum Repeater Networks

Abstract: 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.

• The paper introduces a recursive-bisection tournament that randomly selects among edge-disjoint repeater paths using a tunable bias toward shorter or longer routes.
• Simulations and an analytical model show that an intermediate bias near γ = 1/2 usually outperforms deterministic routing across traffic loads, distance, attenuation, swapping noise, and congestion.
• The balanced policy improves multi-user scalability and Jain fairness while reaching about 86% of a capacity upper bound at low load, though purification and memory decoherence remain untested.

Overview and motivation

The paper addresses a practical control problem in quantum repeater networks: how to select among multiple edge-disjoint repeater paths when distributing entanglement between source–destination (S–D) pairs under lossy links, finite per-link buffer capacities, and probabilistic Bell-state measurements (BSMs). Existing multipath schemes typically rely on centralized controllers performing global optimization—flow formulations or integer programs—which are difficult to execute at the timescales demanded by dynamic traffic. The authors propose instead a lightweight stochastic rule: each entanglement request is routed along one of the available edge-disjoint paths chosen by a randomized "tournament" whose single parameter $\gamma$ biases selection toward shorter or longer routes. The central finding is that an intermediate bias, $\gamma^\ast \approx 1/2$, consistently outperforms both deterministic extremes across distances, loads, attenuation levels, swapping noise, and congestion—a network-level realization of Parrondo-type mixing, where alternating between individually inferior strategies yields a collective gain.

Network model and routing rule

The network is modeled as a random geometric graph (RGG) of $N$ nodes over the unit square, with edges present when node separation $L_{ij} \le r$. Each edge carries transmissivity $p_{\mathrm{link}}^{(ij)} = e^{-\alpha L_{ij}}$ and supports up to $C_0$ entanglement generation attempts per window, so the realized link capacity is binomial, $$C_{ij} \sim \mathrm{Binomial}(C_0, p_{\mathrm{link}}^{(ij)})$$. Generated pairs are assumed high-fidelity and immediately usable; no purification or error correction is modeled. Entanglement extension proceeds via probabilistic BSM-based swapping with success probability $p_{\mathrm{swap}}$, giving a path of hop count $h_i$ a swap weight $w_i = p_{\mathrm{swap}}^{\,h_i - 1}$.

At each time window, $\gamma^\ast \approx 1/2$0 requests are generated for one S–D pair. Feasible edge-disjoint paths are ordered by nondecreasing hop count, and each request is routed by recursive bisection: at every branching level the left (shorter-path) subset is chosen with probability $\gamma^\ast \approx 1/2$1, the right with probability $\gamma^\ast \approx 1/2$2. A request is accepted only if every link on the selected path retains residual capacity; otherwise it is dropped. Throughput is the sum of swap weights over accepted requests.

Analytic throughput model

A key contribution is a closed-form expression for the expected throughput. Conditioning on the number of feasible paths $\gamma^\ast \approx 1/2$3, the tournament induces per-path selection probabilities $\gamma^\ast \approx 1/2$4, where $\gamma^\ast \approx 1/2$5 is the leaf depth and $\gamma^\ast \approx 1/2$6 the number of left turns. Selection counts are binomial in $\gamma^\ast \approx 1/2$7, and because path availability is limited by its bottleneck hop, the effective capacity is $\gamma^\ast \approx 1/2$8, whose mass function follows from products of binomial tail probabilities. This yields

$\gamma^\ast \approx 1/2$9

with $N$0 expressed as a tail sum. Setting the derivative to zero produces an implicit fixed-point equation for the optimal bias $N$1, solved numerically using fitted hop profiles ($N$2, with $N$3 against simulation) and the empirical distribution of $N$4. Analytical predictions agree closely with large-scale simulations averaged over 1000 windows.

Interior optimum and parameter dependence

Simulations reveal a pronounced interior maximum of $N$5 near $N$6 that exceeds both deterministic extremes for all tested request loads. The mechanism is capacity-mediated: mixing distributes traffic across disjoint routes and delays saturation of the bottleneck capacities $N$7, whereas extreme policies concentrate load and exhaust bottlenecks earlier. For light load ($N$8), the optimum drifts toward larger $N$9—a shortest-path preference driven by the strong suppression of long paths when $L_{ij} \le r$0 is small—but for $L_{ij} \le r$1 the optimal bias concentrates tightly near one half, forming a broad plateau where $L_{ij} \le r$2 is small. Heat maps over the $L_{ij} \le r$3 plane confirm that attenuation shifts the optimum only mildly; load is the dominant factor.

Distance scaling reinforces this conclusion. Fitting $L_{ij} \le r$4, the intermediate bias simultaneously maximizes the prefactor $L_{ij} \le r$5 and minimizes the effective attenuation $L_{ij} \le r$6, i.e., it achieves the slowest exponential decay of throughput with distance, while both deterministic extremes degrade faster.

Multi-request performance and fairness

Extending to $L_{ij} \le r$7 concurrent S–D pairs, throughput remains unimodal in $L_{ij} \le r$8 with a maximum near $L_{ij} \le r$9, and aggregate throughput grows approximately linearly in $p_{\mathrm{link}}^{(ij)} = e^{-\alpha L_{ij}}$0—with the largest slope at the balanced bias and substantially smaller slopes at the extremes. This indicates that the Parrondo-type policy scales to multi-user settings sharing a finite entanglement pool. Complementary Jain-index analysis shows that fairness $p_{\mathrm{link}}^{(ij)} = e^{-\alpha L_{ij}}$1 also peaks broadly near $p_{\mathrm{link}}^{(ij)} = e^{-\alpha L_{ij}}$2 and drops toward concentration on few routes at the extremes, with higher loads lifting the entire fairness profile by activating deeper-ranked paths.

Bounds and efficiency

The supplementary material establishes capacity ceilings independent of and dependent on $p_{\mathrm{link}}^{(ij)} = e^{-\alpha L_{ij}}$3: $p_{\mathrm{link}}^{(ij)} = e^{-\alpha L_{ij}}$4, plus a $p_{\mathrm{link}}^{(ij)} = e^{-\alpha L_{ij}}$5-refined envelope. Against these bounds, the tournament policy at optimal bias achieves an efficiency $p_{\mathrm{link}}^{(ij)} = e^{-\alpha L_{ij}}$6 of approximately 0.86 at low load, dipping mid-range as the bound steepens, then recovering and stabilizing at high load as both curves approach the capacity plateau. Structural bounds on $p_{\mathrm{link}}^{(ij)} = e^{-\alpha L_{ij}}$7 follow from the fixed-point relation: universally, $p_{\mathrm{link}}^{(ij)} = e^{-\alpha L_{ij}}$8; in the symmetric equal-depth case $p_{\mathrm{link}}^{(ij)} = e^{-\alpha L_{ij}}$9 exactly; and with depth cap $C_0$0, $C_0$1 lies within $C_0$2. These results explain why the optimum concentrates near one half and quantify how depth asymmetries shift it.

Limitations and open questions

Several assumptions constrain the scope of the results. The model presumes high-fidelity entangled pairs requiring no purification or error correction, which excludes fidelity-aware routing and memory decoherence during storage windows. Link capacities are treated as independent binomial variables within a window, and the analytic expression relies on fitted hop profiles and geometry-averaged link success rather than exact topology-dependent quantities. The fixed-point equation for $C_0$3 is implicit and must be solved numerically, so fully closed-form optimality is not achieved. The paper also leaves open how the tournament policy performs against adaptive or learning-based routing baselines, how it extends to networks with quantum memory constraints across windows, and whether the near-half optimum persists under purification overheads or heterogeneous hardware.

Conclusion

This paper demonstrates that a single-parameter stochastic multipath rule—an ordered recursive-bisection tournament over edge-disjoint paths—achieves robustly near-optimal swap-weighted throughput in noisy quantum repeater networks, outperforming deterministic shortest- and longest-path extremes across loads, distances, attenuation, and swapping noise, while improving Jain fairness and approaching simple capacity upper bounds (efficiency ≈ 0.86 at low load). The accompanying analytic framework connects per-window link capacities to end-to-end rates and yields structural bounds locating the optimal bias near one half. Because the policy requires no global state and minimal classical overhead, it constitutes a practical candidate control layer for near-term repeater networks, though its advantages under purification requirements, memory decoherence, and adaptive baselines remain to be established.