Gottesman-Knill Limit on One-way Communication Complexity: Tracing the Quantum Advantage down to Magic Resources

TL;DR

This paper proves that the quantum advantage in one-way communication complexity with prime-dimensional systems relies on non-stabilizer 'magic' resources, and that stabilizer-based protocols can be classically simulated.

Contribution

It establishes that non-stabilizer resources are essential for quantum advantage in one-way communication, linking it to the Gottesman-Knill theorem.

Findings

Stabilizer-state protocols can be classically simulated without magic resources.

Non-stabilizer 'magic' resources are necessary for quantum advantage.

Minimal magic resources can achieve quantum advantage in specific tasks.

Abstract

Quantum systems are known to offer advantages over their classical counterpart in communication complexity protocols, where the aim is to minimize the amount of information exchange between distant parties to compute global functions of their distributed inputs. In this work, we establish that any one-way communication protocol implemented using a prime-dimensional quantum system -- restricted to stabilizer-state encodings and Clifford-operation decodings -- can be exactly simulated by transmitting a classical system of the same dimension, given access to shared randomness between the sender and receiver. In direct analogy with the Gottesman-Knill theorem, which attributes quantum computational speedup to non-stabilizer resources, commonly known as the magic resources, our result identifies the same non-stabilizer resources as the essential ingredient for the quantum advantage in one-way communication complexity. Furthermore, we present explicit tasks where even a 'minimal magic resource' suffices to achieve a provable quantum advantage, highlighting its efficient use in communication protocols.