Entanglement Routing over Networks with Time Multiplexed Repeaters

Quantum networks will be able to service consumers with long-distance entanglement by use of quantum repeaters that generate Bell pairs (or links) with their neighbors, iid with probability $p$ and perform Bell State Measurements (BSMs) on the links that succeed iid with probability $q$. While global link state knowledge is required to maximize the rate of entanglement generation between any two consumers, it increases the protocol latency due to the classical communication requirements and requires long quantum memory coherence times. We propose two entanglement routing protocols that require only local link state knowledge to relax the quantum memory coherence time requirements and reduce the protocol latency. These protocols utilize multi-path routing protocol and time multiplexed repeaters. The time multiplexed repeaters first generate links for $k$-time steps before performing BSMs on any pairs of links. Our two protocols differ in the decision rule used for performing BSMs at the repeater: the first being a static path based routing protocol and second a dynamic distance based routing protocol. The performance of these protocols depends on the quantum network topology and the consumers' location. We observe that the average entanglement rate and the latency increase with the time multiplexing block length, $k$, irrespective of the protocol. When a step function memory decoherence model is introduced such that qubits are held in the quantum memory for an exponentially distributed time with mean $\mu$, an optimal $k$ ($k_\text{opt}$) value appears, such that for increasing $k$ beyond $k_{\rm opt}$ hurts the entanglement rate. $k_{\rm opt}$ decreases with $p$ and increases with $\mu$. $k_{\rm opt}$ appears due to the tradeoff between benefits from time multiplexing and the increased likelihood of previously established Bell pairs decohering due to finite memory coherence times.