Non-trivial lower bound for 3-coloring the ring in the quantum LOCAL model
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We consider the LOCAL model of distributed computing, where in a single round of communication each node can send to each of its neighbors a message of an arbitrary size. It is know that, classically, the round complexity of 3-coloring an n-node ring is Θ(log^*n). In the case where communication is quantum, only trivial bounds were known: at least some communication must take place. We study distributed algorithms for coloring the ring that perform only a single round of one-way communication. Classically, such limited communication is already known to reduce the number of required colors from Θ(n), when there is no communication, to Θ(log n). In this work, we show that the probability of any quantum single-round one-way distributed algorithm to output a proper 3-coloring is exponentially small in n.
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