Noncommutative Bohnenblust--Hille inequalities

TL;DR

This paper proves a dimension-free Bohnenblust--Hille inequality for quantum Boolean cubes with exponential growth in degree, leading to new results in quantum learning theory and properties of quantum Boolean functions.

Contribution

It provides a new proof of quantum Bohnenblust--Hille inequalities with explicit exponential growth constants, and applies these results to quantum junta theorems and learning problems.

Findings

Dimension-free Bohnenblust--Hille inequalities for quantum Boolean cubes established.

Exponential growth constants in degree for these inequalities derived.

Applications to quantum junta theorems and quantum learning problems demonstrated.

Abstract

Bohnenblust--Hille inequalities for Boolean cubes have been proven with dimension-free constants that grow subexponentially in the degree \cite{defant2019fourier}. Such inequalities have found great applications in learning low-degree Boolean functions \cite{eskenazis2022learning}. Motivated by learning quantum observables, a qubit analogue of Bohnenblust--Hille inequality for Boolean cubes was recently conjectured in \cite{RWZ22}. The conjecture was resolved in \cite{CHP}. In this paper, we give a new proof of these Bohnenblust--Hille inequalities for qubit system with constants that are dimension-free and of exponential growth in the degree. As a consequence, we obtain a junta theorem for low-degree polynomials. Using similar ideas, we also study learning problems of low degree quantum observables and Bohr's radius phenomenon on quantum Boolean cubes.