Quantum learning via harmonic analysis on Boolean cube and cyclic groups

The interaction between learning theory and harmonic analysis was emphasized by mathematics of quantum computing. One of the outstanding open problems in this area concerns the Bohnenblust--Hille inequality
that generalizes a celebrated Littlewood’s 4/3 lemma. How to learn (approximately and with large probability) a very large matrix in a relatively small number of random quantum quarries? Motivated by this question, a non-commutative counterpart of Bohnenblust--Hille inequality for Boolean cubes was recently conjectured in Cambyse Rouz, Melchior Wirth, and Haonan Zhang. By waving the hands I will explain the proof of non-commutative Bohnenblust--Hille inequalities with constants that are dimension-free. As applications, we study learning problems of quantum observables. Using Heisenberg—Weyl basis one
reduces the quantum problem to commutative problem: the Bohnenblust—Hille inequality for cyclic groups (joint with Haonan Zhang, Joe Slote). To prove the Bohnenblust—Hille inequality for cyclic groups turned out to be a challenging problem. I will explain the progress in this area.