Distributed Quantum Algorithms Cannot Color Cycles with Probability 1

2026-08-12

AI Summary

This work investigates whether distributed quantum algorithms can efficiently solve the 3-coloring problem on anonymous ring graphs with probability one. By introducing a novel “truly quantum” lower-bound technique—capable for the first time of distinguishing distributions permitted by physical causality from those achievable by quantum protocols—and combining symmetry-breaking arguments with a wishful quantum teleportation strategy, the authors reduce any T-round algorithm to an equivalent single-round protocol. This reduction establishes that any such quantum algorithm requires Ω(n) rounds of communication and must be inherently global. The result thus demonstrates a fundamental limitation of quantum algorithms for this problem, conclusively ruling out the possibility of an efficient quantum solution and showing that quantum approaches cannot surpass classical ones in this setting.

Abstract

We prove that any distributed quantum algorithm that finds a $3$-coloring with probability $1$ in a cycle of anonymous identical computers has to be global, that is, it needs $Ω(n)$ communication rounds. It follows that quantum computation and communication does not help with this problem. All prior lower bounds on quantum advantage in distributed graph algorithms use arguments related to physical causality. However, it is known that such arguments cannot rule out fast quantum advantage for $3$-coloring cycles. In particular, any causality-based argument would rule out the existence of finitely dependent coloring, but Holroyd and Liggett (2016) showed that such colorings do exist. Hence to tackle this problem, we need a ``genuinely quantum'' lower-bound technique that can distinguish between (1) distributions that do not violate physical causality vs. (2) distributions that can be realized with a quantum strategy. We present the first such lower-bound technique in this context. First, we show that $1$-round quantum algorithms cannot break symmetry with probability $1$. Second, we present a wishful teleportation strategy that can be used to turn $T$-round quantum 3-coloring algorithms into $1$-round quantum algorithms breaking symmetry, while preserving success probability $1$. Put together, the lower bound follows.

Problem

• distributed quantum algorithms
• 3-coloring
• cycle graph
• quantum advantage
• symmetry breaking

Innovation

• distributed quantum algorithms
• quantum lower bounds
• graph coloring
• symmetry breaking
• non-causal quantum strategies