Clean quantum and classical communication protocols

TL;DR

This paper introduces the concept of clean communication protocols in quantum and classical settings, providing efficient protocols for the Inner Product function and analyzing communication costs for general functions.

Contribution

It presents the first clean protocols for the Inner Product function and establishes near-maximal communication bounds for general Boolean functions with and without entanglement.

Findings

Quantum clean protocol for Inner Product requires at most n+3 qubits.

Classical clean protocol for Inner Product requires at most n+O(√n) bits.

Most Boolean functions need close to 2n bits or n qubits of communication.

Abstract

By how much must the communication complexity of a function increase if we demand that the parties not only correctly compute the function but also return all registers (other than the one containing the answer) to their initial states at the end of the communication protocol? Protocols that achieve this are referred to as clean and the associated cost as the clean communication complexity. Here we present clean protocols for calculating the Inner Product of two $n$-bit strings, showing that (in the absence of pre-shared entanglement) at most $n+3$ qubits or $n+O(\sqrt{n})$ bits of communication are required. The quantum protocol provides inspiration for obtaining the optimal method to implement distributed CNOT gates in parallel whilst minimizing the amount of quantum communication. For more general functions, we show that nearly all Boolean functions require close to $2n$ bits of classical communication to compute and close to $n$ qubits if the parties have access to pre-shared entanglement. Both of these values are maximal for their respective paradigms.