Limitations of nearest-neighbor quantum networks

Quantum communication research has in recent years shifted to include multipartite networks for which questions of quantum network routing naturally emerge. To understand the potential for multipartite routing, we focus on the most promising architectures for future quantum networks—those connecting nodes close to each other. Nearest-neighbor networks, such as rings, lines, and grids, have been studied under different communication scenarios to facilitate the sharing of quantum resources especially in the presence of bottlenecks. We analyze the potential of nearest-neighbor entangling gate quantum networks and identify some serious limitations by demonstrating that rings and lines cannot overcome common bottleneck communication problems.

Figure 1

Measurements on a ring. From left to right: Ring $R_{n+1}$ with $n+1$ vertices, node $v$ is measured in the $Z$, $Y$, and $X$ bases.

Figure 2

The foliage is LC invariant. The arrows indicate how foliage nodes $u,w$ transform under LCs. The central four graphs show twins $v,w$ that can be (dis-)connected via a LC with respect to one of their neighbors $u$. The LCs ${\tau }_v$ and ${\tau }_w$ transform the outer graphs with the connected twins into graphs where either $v$ (upper row) or $w$ (lower row) is an axil and their former twin partner a leaf.

Figure 3

Ring (a) and butterfly (e) network. Whereas ring and butterfly networks are not LC equivalent, a ring graph (a) can be transformed via LCs into a butterflylike graph (c) where two nodes, here 1 and 2, are swapped ($\widehat{=}$). The ring $R_6$ is transformed via a local complementation sequence (a), $\left(b\right)\widehat{=}\left(c\right)$. From the resulting graph (c), one can obtain maximally entangled pairs (d) between nodes (2,4) and (1,5) by measuring 3 and 6. When allowing for two-local operations (specifically $CZ$ gates) between nodes 3 and 6 (e) one can obtain crossing Bell pairs (1,4) and (2,5) (f).