A Dynamic Multiplexing Policy for a Quantum Repeater

Technical Summary: A Dynamic Multiplexing Policy for a Quantum Repeater

Problem Statement
The realization of large-scale quantum networks is currently bottlenecked by the requirement that the rate of entanglement generation must exceed the decoherence rate of entanglement stored in quantum memories. While multiplexing (utilizing temporal, spectral, or spatial modes) can improve generation rates, most existing analytical work on multiplexed quantum repeaters assumes a fixed assignment of quantum chips (modes) to specific end nodes. This paper addresses the potential benefits of dynamic multiplexing, where a reconfigurable optical router allows the assignment of quantum chips to end nodes to change on a cycle-by-cycle basis. The authors specifically investigate a regime relevant to near-term quantum networks, characterized by low entanglement generation probabilities (order of $10^{-5}$ to $10^{-6}$) and limited memory coherence times, where average link generation per cycle is less than one.

Methodology
The authors model a multiplexed quantum repeater node connecting two end nodes (left and right) using $2m$ quantum chips. Each chip contains a communication qubit and a memory qubit. The system operates in discrete time steps comprising synchronization, remote entanglement generation, and optional entanglement swapping phases.

1. Policies Compared:

   • Fixed Multiplexing (FxdMux): A static assignment where $m$ chips are permanently dedicated to the left end node and $m$ to the right.
   • Dynamic Multiplexing (DynMux): A policy where, if an entangled link is successfully generated with one end node, all remaining free chips are immediately assigned to the opposite end node in the next cycle. This aims to minimize the time a link waits in memory before a matching link is generated for swapping.

2. Analytical Framework:

   • The system state is modeled as a Markov chain representing the signed queue length (difference between links to the left and right nodes).
   • Exact Analysis: The authors derive exact expressions for the steady-state rate and Werner parameter (a proxy for fidelity) by solving linear systems of size $O(m)$ using the Ergodic Theorem for Markov-modulated processes.
   • Approximate Analysis: A "One-Simultaneous-Success" (OSS) approximation is introduced, assuming at most one link is generated per time step (valid for near-term parameters). This yields closed-form expressions for performance metrics, facilitating heuristic optimization of the secret key rate.
   • Noise Model: The analysis employs a depolarizing noise model to account for time-dependent decoherence in memory qubits and static noise from imperfect operations.
   • Validation: The analytical results are validated against discrete-event simulations using the NetSquid framework.

3. Hardware Considerations:

   • The paper explicitly models the optical router depth. DynMux requires a deeper router ($d_{dyn} = 2\log_2(m) + 1$) compared to FxdMux ($d_{fxd} = \log_2(m)$), introducing higher switching losses.

Key Contributions

• Analytical Comparison: The paper provides the first exact Markov chain analysis comparing dynamic and fixed multiplexing policies for a symmetrically placed repeater, quantifying improvements in steady-state rate, fidelity, and secret key rate.
• Fidelity Improvement Mechanism: The authors demonstrate that DynMux improves fidelity not only by reducing the waiting time for the first link in the queue but also by preventing the accumulation of multiple links for the same end node (queuing), which would otherwise suffer significant decoherence while waiting for a match.
• Secret Key Rate Heuristic: The study evaluates the trade-off between rate and fidelity via the secret key rate, optimizing over the cutoff for local entanglement generation attempts.

Results

• Rate vs. Fidelity: In the near-term regime (low generation probability, short coherence times), DynMux yields a marginally higher rate but a significantly higher fidelity (Werner parameter) compared to FxdMux. For example, with 512 chips and 10s coherence time, DynMux achieves a Werner parameter near 1, whereas FxdMux remains around 0.6.
• Secret Key Rate:
  • With lossless switches, DynMux achieves positive secret key rates with fewer chips (starting at $\log_2(m)=2$) compared to FxdMux (starting at $\log_2(m)=4$).
  • The ratio of secret key rates (DynMux/FxdMux) is highest for small numbers of chips (e.g., 23.9x higher with 4 chips) and decreases as the number of chips increases (1.65x with 512 chips).
  • Switching Losses: Even when accounting for switching inefficiencies (e.g., 90% efficiency per switch), DynMux maintains a positive secret key rate in regimes where FxdMux drops to zero. This holds true despite DynMux requiring a deeper, more lossy router.

Significance and Claims
The paper claims that dynamic multiplexing is especially relevant for near-term quantum networks where the number of available quantum chips is limited and entanglement generation probabilities are low. By dynamically reassigning resources, the policy mitigates the decoherence of stored entanglement more effectively than fixed assignments, leading to substantial improvements in fidelity and secret key rates. The authors note that while their specific policy is not proven optimal, it demonstrates that dynamic resource assignment can outperform fixed strategies even in the presence of the additional optical losses inherent in more complex reconfigurable routers. This suggests that the development of reconfigurable routers for many-chip repeaters is a viable and advantageous path forward for near-term quantum network deployment.