Exponential Quantum Advantage in Numbers-on-Forehead Communication

2026-09-30

AI Summary

This study addresses the long-standing absence of exponential quantum advantages for general interactive protocols in the three-party Number-On-the-Forehead (NOF) model, where existing results are largely confined to one-way communication. To overcome this limitation, the authors construct an interleaved unitary product problem and perform complexity analysis by integrating regularity decomposition, matrix product estimation, and a novel randomized lower bound argument. This work establishes the first exponential quantum advantage in the general interactive NOF model, surpassing previous one-way restrictions. Specifically, the proposed function requires only O(log n) qubits of quantum communication, whereas its classical randomized communication complexity admits a lower bound of Ω(n^{1/32}). This polynomial separation rigorously demonstrates the substantial superiority of quantum interactive protocols over their classical counterparts in multiparty communication complexity.

Abstract

We give the first exponential quantum advantage in the general interactive three-party Numbers-on-Forehead (NOF) model for a decision problem. Previous separations hold only for restricted protocols like one-way communication for a relation. We construct an explicit partial Boolean function, the Interleaved Unitary Product problem, that requires only $O(\log n)$ NOF quantum communication but $\widetildeΩ(n^{1/32})$ randomized communication. This function builds on the two-party unitary product problem of Arunachalam, Girish, and Lifshitz (TQC 2024). The main technical obstacle is that discrepancy, the standard lower-bound method for NOF, also lower-bounds quantum communication. We instead develop a regularity-based argument for randomized NOF lower bounds, building on the approach of Kelley, Lovett, and Meka and adapting the regularity decomposition of Abboud, Fischer, Kelley, Lovett, and Meka (STOC 2024) to cylinder intersections. Combined with matrix-product estimates of Arunachalam, Girish, and Lifshitz, this yields our randomized lower bound.

Problem

• Numbers-on-Forehead
• Quantum communication complexity
• Exponential quantum advantage
• Randomized communication
• Decision problem

Innovation

• Numbers-on-Forehead model
• Exponential quantum advantage
• Regularity-based argument
• Cylinder intersections
• Interleaved Unitary Product problem