Mind the O˜: Asymptotically Better, but Still Impractical, Quantum Distributed Algorithms.

We present two algorithms in the quantum CONGEST-CLIQUE model of distributed computation that succeed with high probability: one for producing an approximately optimal Steiner tree, and one for producing an exact directed minimum spanning tree, each of which uses O˜(n1/4) rounds of communication and O˜(n9/4) messages, achieving a lower asymptotic round and message complexity than any known algorithms in the classical CONGEST-CLIQUE model. At a high level, we achieve these results by combining classical algorithmic frameworks with quantum subroutines. Additionally, we characterize the constants and logarithmic factors involved in our algorithms as well as related classical algorithms, otherwise obscured by O˜ notation, revealing that advances are needed to render both the quantum and classical algorithms practical.