Congestion-free Multiflow Quantum Tree Network with Logarithmic Overhead

We propose quantum tree network (QTN) designs that are 1) free from congestion in multiflow communications, 2) optimal routing with only local information (self-information) and 3) completely covers 2D surface, 4) with only logarithmic overhead. If the channel is dominated by the insertion loss and the communication is limited by the local gates, as in current state-of-the-art technologies, the overhead is log(N), while this is relaxed to the general cases with poly-log overhead by engaging repeater chains in the long range communication.