Entanglement Routing and Bottlenecks in Grid Networks

TL;DR

This paper investigates entanglement routing in quantum grid networks, revealing that optimal paths for protocols may not be shortest, and explores bottlenecks affecting Bell pair distribution in various network topologies.

Contribution

It extends existing graph-theoretic protocols to show non-shortest paths can be more optimal and analyzes bottleneck issues in grid and other network architectures.

Findings

Optimal entanglement routing may not follow shortest paths.

Bottlenecks significantly impact Bell pair distribution in grid networks.

Line and ring networks are also prone to bottlenecks, affecting quantum communication.

Abstract

Distributing entangled pairs among multiple users is a fundamental problem in quantum networks. Existing protocols like $X$ protocol introduced in (npj Quantum Information 5, 76 (2019)) use graph theoretic tools like local complementation to optimize the number of measurements required to extract any Bell pair among the network users. However, such a protocol relies on finding the shortest path between the users. Here, the existing results are extended to establish a counter-intuitive notion that, in general, the most optimal path to perform the $X$ protocol is not along the shortest path. Specific examples of this advantage are provided on networks of size as small as 12 qubits. Bottlenecks in establishing simultaneous Bell pairs in nearest-neighbor architectures are also explored. Recent results suggesting the unsuitability of the line and ring networks for the implementation of quantum networks due to the existence of bottlenecks are revisited, and using local equivalency relations from graph theory, it is hinted at the possibility that even grid graphs are not exempt from bottleneck issues. Further, it's noted that the results obtained here would be of use in analyzing the advantages of measurement-based quantum network coding.