Quantum physics is known to allow for completely new ways to create, manipulate and store information. Quantum communication--the ability to transmit quantum information--is a primitive necessary for any quantum internet. At its core, quantum communication generally requires the formation of entangled links between remote locations. The performance of these links is limited by the classical signalling time between such locations, necessitating the need for long-lived quantum memories. Here, we present the design of a communications network that neither requires the establishment of entanglement between remote locations nor the use of long-lived quantum memories. The rate at which quantum data can be transmitted along the network is only limited by the time required to perform efficient local gate operations. Our scheme therefore has the potential to provide higher communications rates than previously thought possible.