Non-trivial lower bound for 3-coloring the ring in the quantum LOCAL model

TL;DR

This paper proves that in the quantum LOCAL model, a single-round one-way distributed algorithm cannot efficiently 3-color a ring, with the success probability decreasing exponentially with the number of nodes.

Contribution

It establishes a non-trivial lower bound on the success probability of quantum algorithms for 3-coloring in the LOCAL model with limited communication.

Findings

Quantum single-round algorithms have exponentially small success probability for 3-coloring.

Classical limited communication reduces coloring to logarithmic colors, but quantum does not significantly improve this.

The result highlights fundamental limitations of quantum communication in distributed graph coloring.

Abstract

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 $\Theta(\log^*\!n)$. 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 $\Theta(n)$, when there is no communication, to $\Theta(\log n)$. 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 $n$.