 | Seminario Matematico e Fisico di Milano |
| Alexander Volberg
Michigan State University
Quantum learning via harmonic analysis on Boolean cube and cyclic groups
Tuesday, June 13 2023, at 16:30 precise
Sala di Rappresentanza, Dipartimento di Matematica, Università di Milano |
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Abstract
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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.
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