Quantum-Classical Equivalence for AND-Functions

2026-06-02

AI Summary

This work investigates whether quantum communication protocols can achieve exponential advantage over classical ones for total Boolean functions, focusing on the communication complexity of generalized AND-functions—compositions of an arbitrary Boolean function \( f \) with the two-bit AND gate (\( f \circ \text{AND}_2 \)). By extending Razborov’s result on symmetric outer functions to arbitrary Boolean functions and combining it with a logarithmic characterization based on De Morgan sparsity alongside the structural analysis of non-sparse functions by Chattopadhyay et al., the paper establishes that for any \( f \), the bounded-error quantum communication complexity of \( f \circ \text{AND}_2 \) is polynomially equivalent—up to polylogarithmic factors—to its classical deterministic communication complexity. This resolves a long-standing conjecture by confirming polynomial equivalence between quantum and classical communication complexities for generalized AND-functions.

Abstract

A major open problem in quantum communication complexity is whether quantum protocols can be exponentially more efficient than classical protocols for computing total Boolean functions; the prevailing conjecture is that they cannot be so. In a seminal work, Razborov (2002) resolved this question for AND-functions of the form $$ F(x,y) = f(x_1 \land y_1, \ldots, x_n \land y_n), $$ when the outer function $f$ is symmetric, by proving that their bounded-error quantum and classical communication complexities are polynomially related. Since then, extending this result to all AND-functions has remained open and has been posed by several authors. In this work, we settle this problem in a strong way. We show that for every Boolean function $f$, the bounded-error quantum and classical deterministic communication complexities of the function $f \circ \mathrm{AND}_2$ are polynomially related, up to polylogarithmic factors in $n$. We prove this by showing that both are characterized--up to polynomial loss--by the logarithm of the De Morgan sparsity of $f$. Our results build on the recent work of Chattopadhyay, Dahiya, and Lovett (2025) on structural characterizations of non-sparse Boolean functions, which we extend to resolve the conjecture for general AND-functions.

Problem

• quantum communication complexity
• classical communication complexity
• AND-functions
• total Boolean functions
• polynomial equivalence

Innovation

• quantum communication complexity
• AND-functions
• De Morgan sparsity
• classical-quantum equivalence
• Boolean functions