On Quantum-Classical Equivalence for Composed Communication Problems

TL;DR

This paper demonstrates that for any Boolean function, the quantum and classical complexities of computing both the AND and OR of inputs are polynomially related, extending previous results and connecting quantum complexity to classical measures.

Contribution

It proves that computing both f(x AND y) and f(x OR y) has polynomially related classical and quantum complexities for all Boolean functions, addressing a variant of an open problem.

Findings

Quantum and classical complexities are polynomially related for computing both AND and OR of inputs.

Quantum complexity is polynomially related to classical deterministic complexity and block sensitivity.

Result holds regardless of prior entanglement.

Abstract

An open problem in communication complexity proposed by several authors is to prove that for every Boolean function f, the task of computing f(x AND y) has polynomially related classical and quantum bounded-error complexities. We solve a variant of this question. For every f, we prove that the task of computing, on input x and y, both of the quantities f(x AND y) and f(x OR y) has polynomially related classical and quantum bounded-error complexities. We further show that the quantum bounded-error complexity is polynomially related to the classical deterministic complexity and the block sensitivity of f. This result holds regardless of prior entanglement.