Optimal and Verifiable Quantum Advantages in Communication Complexity

2026-10-04

AI Summary

This study addresses the longstanding absence of efficient verifiability in optimal quantum-versus-classical separations within communication complexity. To bridge this gap, it proposes search problems such as Pelagic Fourier Fishing, which integrate quantum one-way communication protocols, Clifford hierarchy unitaries, and stabilizer state measurements to construct a family of total search problems whose solutions are verifiable in polynomial time. This work achieves the first optimal and efficiently verifiable quantum advantage, establishing exponential separations between n qubits of quantum communication and Ω(2^n) as well as Ω_d(n^d) classical communication. Consequently, it provides an unconditional framework for verifiable quantum advantage applicable to near-term quantum experiments.

Abstract

We establish optimal quantum-classical separations in communication complexity for search problems. We introduce a total search problem called Pelagic Fourier Fishing and show that it admits an $n$-qubit quantum one-way protocol, whereas every randomized two-way protocol requires $\Omega(2^n)$ bits of communication. We then introduce a variant of this problem whose solutions can be verified in polynomial time. This variant also admits an $n$-qubit quantum one-way protocol, while every randomized one-way protocol requires $\Omega(2^n)$ bits of communication. We also construct a family of efficiently verifiable total search problems achieving an $n$ versus $\Omega_d(n^d)$ separation between quantum one-way and randomized one-way communication for every fixed $d \ge 2$. In the quantum protocol, Alice prepares her message using a single unitary from the $d$th level of the Clifford hierarchy, and Bob performs a Clifford measurement. This separation is asymptotically optimal under this restriction on Alice's message. For $d = 2$, Alice's message is a stabilizer state and Bob's measurement is Clifford, so the protocol uses no magic, yet achieves an optimal quadratic quantum advantage. Finally, we discuss how these separations can be adapted to near-term quantum advantage experiments in which the demonstrated advantage is both unconditional and efficiently verifiable.

Problem

• communication complexity
• quantum advantage
• search problems
• verifiability
• quantum-classical separation

Innovation

• Communication Complexity
• Quantum Advantage
• Search Problems
• Clifford Hierarchy
• Verifiable Quantum Protocols