Scalable Quantum Networks: Congestion-Free Hierarchical Entanglement Routing with Error Correction

We introduce Quantum Tree Networks (QTN), an architecture for hierarchical multi-flow entanglement routing. The network design is a $k$-ary tree where end nodes are situated on the leaves and routers at the internal nodes, with each node connected to $k$ nodes in the child layer. The channel length between nodes grows with a rate $a_k$, increasing as one ascends from the leaf to the root node. This construction allows for congestion-free and error-corrected operation with qubit-per-node overhead to scale sublinearly with the number of end nodes, $N$. The overhead for a $k$-ary QTN scales as $\mathcal{O}(N^{\log_k a_k} \cdot \log_k N)$ and is sublinear for all $k$ with minimal surface-covering end nodes. More specifically, the overhead of quarternary ($k=4$) QTN is $\sim \mathcal{O}(N^{0.25}\cdot\log_4 N)$. Alternatively, when end nodes are distributed over a square lattice, the quaternary tree routing gives the overhead $\sim \mathcal{O}(\sqrt{N}\cdot\log_4 N)$. Our network-level simulations demonstrate a size-independent threshold behavior of QTNs. Moreover, tree network routing avoids the necessity for intricate multi-path finding algorithms, streamlining the network operation. With these properties, the QTN architecture satisfies crucial requirements for scalable quantum networks.