One Clean Qubit Suffices for Quantum Communication Advantage

TL;DR

This paper proves that even with only one clean qubit, quantum communication can outperform classical methods significantly, establishing new bounds and separations in quantum communication complexity.

Contribution

It demonstrates a quantum protocol with logarithmic cost using one clean qubit, contrasting with classical protocols requiring square root cost, and employs advanced mathematical tools for proof.

Findings

Quantum protocol with O(log N) cost using one clean qubit

Classical randomized protocols require at least Ω(√N) cost

Quantum-simultaneous-with-entanglement protocol also efficient at O(log N)

Abstract

We study the one-clean-qubit model of quantum communication where one qubit is in a pure state and all other qubits are maximally mixed. We demonstrate a partial function that has a quantum protocol of cost $O(\log N)$ in this model, however, every interactive randomized protocol has cost $\Omega(\sqrt{N})$, settling a conjecture of Klauck and Lim. In contrast, all prior quantum versus classical communication separations required at least $\Omega(\log N)$ clean qubits. The function demonstrating our separation also has an efficient protocol in the quantum-simultaneous-with-entanglement model of cost $O(\log N )$. We thus recover the state-of-the-art separations between quantum and classical communication complexity. Our proof is based on a recent hypercontractivity inequality introduced by Ellis, Kindler, Lifshitz, and Minzer, in conjunction with tools from the representation theory of compact Lie groups.