Quantum routing in planar graph using perfect state transfer

In this article, we consider a spin-spin interaction network governed by $XX+YY$ Hamiltonian. The vertices and edges of the network represent the spin objects and their interactions, respectively. We take a privilege to switch on or off any interaction, that assists us to perform multiple perfect state transfers in a graph simultaneously. We also build up a salable network allowing quantum communication between two arbitrary vertices. Later we utilize the combinatorial characteristics of hypercube graphs to propose a static routing schema to communicate simultaneously between a set of senders and a set of receivers in a planar network. Our construction is new and significantly powerful. We elaborate multiple examples of planar graphs supporting quantum routing where classical routing is not possible.