Balancing Quantum Memories in Asymmetric Repeaters for High-Fidelity Entanglement Distribution

Abstract: At the core of the quantum Internet lie quantum repeaters that enable remote end-to-end entanglement generation. Fundamentally, the entanglement generation rate and fidelity of quantum repeaters constitute the bottleneck for end-to-end performance. To achieve high rates, quantum repeaters employ quantum memory multiplexing. In a high-rate standard repeater, each memory sequentially generates an entanglement with its neighboring nodes and then applies entanglement swapping. This, however, results in low fidelity due to decoherence of the first-formed entanglement in the sequential generation process. By allocating different numbers of memories to simultaneously form entanglements with the left and right adjacent nodes, quantum repeaters reduce high waiting times and achieve high fidelity. In such a repeater, a mismatch problem arises due to the difference between the probabilistic number of generated entanglements on both sides. Consequently, some entanglements remain stored until opposite entanglements are available. The mismatch problem reduces the repeater rate and particularly the entanglement fidelity. In this paper, we consider the mismatch problem in an asymmetric repeater with different distances to its adjacent nodes. To mitigate the mismatch problem, we derive a dynamic optimal memory allocation. Under the optimal allocation, we derive statistical lower bounds on the achievable rate and fidelity. We demonstrate that the optimal allocation significantly improves the fidelity while maintaining a comparable rate to the standard repeater. In contrast, our results show that fixed memory allocation may be detrimental to the fidelity.

• The paper derives a locally computable dynamic allocation that balances left- and right-link entanglement successes, reducing unmatched memories while approaching the standard repeater rate as memory size grows.
• The analysis shows that rate loss scales as O(√N) against a linear leading term, while mismatch-driven fidelity degradation can fall exponentially, producing fidelities near 0.96 versus about 0.55 with equal allocation at N=100.
• The paper introduces hard cutoffs for allocation violations, finding that they improve fidelity with little rate loss for N≥10, whereas very small memories face a substantial rate–fidelity trade-off.

Overview

This paper by Elsayed and Rizk (Leibniz University Hannover) addresses the mismatch problem in multiplexed quantum repeaters that generate entanglement simultaneously with both adjacent nodes. In such repeaters, the numbers of successful entanglements on the left and right sides are binomially distributed, so some entanglements remain unmatched across rounds and decohere while waiting. The authors consider the asymmetric case—where repeater-to-node distances $d_l \neq d_r$—and derive a dynamic memory allocation that minimizes expected unmatched entanglements, together with statistical lower bounds on rate and fidelity under this allocation (2604.24554).

System model

The repeater holds $N$ memories freely assignable to left or right banks of sizes $N_l(t)$ and $N_r(t)$. Each round consists of an entanglement generation phase (Bernoulli trials with success probabilities $p_l = e^{-\sigma d_l}$ and $p_r = e^{-\sigma d_r}$ over a shared multiplexed channel) followed by local matching and entanglement swapping, enabled by all-to-all connectivity such as a Mach-Zehnder interferometer switch or trapped-ion architectures. The mismatch evolves as $\alpha(t+1) = \alpha(t) + X_l(t) - X_r(t)$, with binomial trial counts adjusted for memories occupied by stored entanglements. Fidelity follows Werner-state depolarization, decaying exponentially with entanglement age relative to the coherence time $t_c$, and swapping combines fidelities as $$F_{\text{swap}} = F_1F_2 + \frac{1}{3}(1-F_1)(1-F_2)$$.

Optimal allocation and performance bounds

Since the median minimizes expected absolute error, the optimal allocation should enforce zero median of $\alpha$. Because the stationary distribution of $N$0 is analytically intractable for arbitrary $N$1, the authors instead impose the zero-mean condition $N$2, which is asymptotically equivalent under the Central Limit Theorem approximation of the binomial counts by Gaussians. The resulting allocation assigns to the right bank

$N$3

with $N$4 if $N$5 and $N$6 otherwise; i.e., bank sizes track both the link asymmetry and the current imbalance. Under this allocation, the matched-entanglement count satisfies

$N$7

so the rate penalty scales as $N$8 against a linear leading term—the bound approaches the standard repeater rate $N$9 asymptotically. Using FIFO matching and the additive drift theorem, the expected age of unmatched entanglements is bounded by $N_l(t)$0, which via Cauchy–Schwarz and Jensen's inequality yields a closed-form lower bound on swapped fidelity. A key implication is that because fidelity decays exponentially in the mismatch while rate loss is only square-root, mismatch minimization buys fidelity at negligible rate cost.

Allocation violation and hard-cutoff regime

The zero-mean condition cannot always be met: when required bank sizes fall below the number of stored unmatched entanglements (e.g., $N_l(t)$1 with $N_l(t)$2), the optimal allocation is violated. The paper identifies explicit violation thresholds, e.g., $N_l(t)$3 for right-side excess, and proposes a hard-cutoff regime that drops entanglements beyond these thresholds. Numerically, for $N_l(t)$4 km, $N_l(t)$5 km, $N_l(t)$6 s, hard-cutoff strictly dominates for $N_l(t)$7: near-identical rate with substantially higher fidelity. For $N_l(t)$8 the threshold shrinks toward zero, cutoffs discard nearly everything, and a genuine rate–fidelity trade-off emerges—an order-of-magnitude rate loss for near-perfect fidelity.

Numerical results

Monte Carlo evaluation ($N_l(t)$9 dB/km, $N_r(t)$0) compares optimal, fixed equal ($N_r(t)$1 per side), fixed proportional ($N_r(t)$2), and standard sequential repeaters. The central findings are:

• Optimal allocation achieves near-standard rates: the relative rate gap to the standard repeater falls from roughly 55% at $N_r(t)$3 to about 1.5% at $N_r(t)$4 for $N_r(t)$5, $N_r(t)$6 km.
• Fixed equal allocation is strongly detrimental: it incurs a nearly constant ~28% rate deficit regardless of $N_r(t)$7, and its fidelity can be worse than the standard repeater (e.g., ~0.55 vs. ~0.52 at large $N_r(t)$8 for one parameterization)—a notable negative result, since equal splitting is the natural default.
• Fidelity gains are large: optimal allocation reaches ~0.96 at $N_r(t)$9 versus ~0.55 for equal allocation and ~0.52 for the standard repeater ($p_l = e^{-\sigma d_l}$0, $p_l = e^{-\sigma d_l}$1 km, $p_l = e^{-\sigma d_l}$2 s). The gap between optimal and proportional allocation widens with longer links and shorter coherence times, consistent with the exponential dependence of fidelity on $p_l = e^{-\sigma d_l}$3.
• Bounds are tight: simulated rate and fidelity under optimal allocation closely track the analytical lower bounds.

Application to repeater chains

Because the allocation requires only local knowledge, it composes directly into multi-repeater chains under arbitrary scheduling protocols. A two-repeater chain simulation ($p_l = e^{-\sigma d_l}$4 km, $p_l = e^{-\sigma d_l}$5 km) reproduces the single-repeater behavior: substantial fidelity gains over equal allocation and the standard chain, with a slightly larger rate loss due to idle memories arising when adjacent repeaters allocate different bank sizes to their shared link. The paper flags this memory underutilization as an open problem, suggesting joint optimization of per-repeater memory counts in heterogeneous systems.

Limitations and open questions

Several assumptions constrain the results. The optimality of the zero-mean condition rests on the CLT-based mean–median equivalence, exact only asymptotically in $p_l = e^{-\sigma d_l}$6; small-$p_l = e^{-\sigma d_l}$7 regimes show elevated rate errors from allocation violations. The rate bound derivation approximates $p_l = e^{-\sigma d_l}$8 as negligible, justified only heuristically by mismatch minimization. Hardware-induced fidelity losses (Bell-state measurements, switching gates) are excluded on the grounds that they are allocation-independent, though the architecture demands more gates than the standard repeater—a cost outweighed by decoherence losses only when link distances are not very short. The asymmetric round duration is assumed not to cause additional per-round fidelity decay. Finally, the multi-repeater analysis covers a single two-repeater greedy example; scaling behavior over many repeaters and interaction with swapping-tree scheduling protocols remain unanalyzed.

Conclusion

The paper provides an analytic, locally computable dynamic memory allocation for asymmetric multiplexed repeaters that provably bounds—and empirically demonstrates—near-standard rates with substantially improved fidelity, while showing that naive fixed allocations can underperform even the sequential standard repeater. The hard-cutoff extension handles finite-memory violation of the optimality condition, dominating for sufficiently large $p_l = e^{-\sigma d_l}$9. The main open questions are memory-count co-design across chained repeaters to eliminate idle memories and integration with heterogeneous scheduling protocols.