Alexander Volberg

Analysis on the Boolean hypercube −1,1^n, particularly Fourier analysis, has played a crucial role in many areas of mathematics and computer science, including learning algorithms. In view of the success and importance of quantum algorithms, it is natural to transfer classical learning results to the quantum realm. In this contribution, we extend the recent progress on learning low-degree functions and quantum operators to the setting of general qudit systems, as well as develop novel Fourier-analytic tools for studying (generalized) Pauli decompositions of Hermitian operators. These tools also allow us to deduce new results in quantum Boolean analysis and approximate theory, such as the junta-type theorem and dimension-free discrete Remez-type inequalities.