Quantum Distributed Complexity of Set Disjointness on a Line

Set Disjointness on a Line is a variant of the Set Disjointness problem in a distributed computing scenario with $d+1$ processors arranged on a path of length $d$. It was introduced by Le Gall and Magniez (PODC 2018) for proving lower bounds on the quantum distributed complexity of computing the diameter of an arbitrary network in the CONGEST model. However, they were only able to provide a lower bound when the local memory used by the processors on the intermediate vertices of the path consists of O$( \log n)$ qubits for $n$-bit inputs. We prove an unconditional lower bound of $\widetildeΩ\big(\sqrt[3]{n d^2}+\sqrt{n} \, \big)$ rounds for Set Disjointness on a Line with $d + 1$ processors. The result gives us a new lower bound of $\widetildeΩ \big( \sqrt[3]{nδ^2}+\sqrt{n} \, \big)$ on the number of rounds required for computing the diameter $δ$ of any $n$-node network with quantum messages of size O$(\log n)$ in the CONGEST model. We draw a connection between the distributed computing scenario above and a new model of query complexity. The information-theoretic technique we use for deriving the round lower bound for Set Disjointness on a Line also applies to the number of rounds in this query model. We provide an algorithm for Set Disjointness in this query model with round complexity that matches the round lower bound stated above, up to a polylogarithmic factor. This presents a barrier for obtaining a better round lower bound for Set Disjointness on the Line. At the same time, it hints at the possibility of better communication protocols for the problem.