Mind the Õ: 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 exact directed minimum spanning each which uses O˜(n1/4) rounds communication O˜(n9/4) messages, achieving a lower asymptotic round message complexity than any known classical model. At level, we achieve these results by combining algorithmic frameworks subroutines. Additionally, characterize constants logarithmic factors involved our as well related algorithms, otherwise obscured O˜ notation, revealing advances are needed to render both practical.