Volberg, Alexander; Zhang, HaonanISTA
Bohnenblust–Hille inequalities for Boolean cubes have been proven with dimension-free constants that grow subexponentially in the degree (Defant et al. in Math Ann 374(1):653–680, 2019). Such inequalities have found great applications in learning low-degree Boolean functions (Eskenazis and Ivanisvili in Proceedings of the 54th annual ACM SIGACT symposium on theory of computing, pp 203–207, 2022). Motivated by learning quantum observables, a qubit analogue of Bohnenblust–Hille inequality for Boolean cubes was recently conjectured in Rouzé et al. (Quantum Talagrand, KKL and Friedgut’s theorems and the learnability of quantum Boolean functions, 2022. arXiv preprint arXiv:2209.07279). The conjecture was resolved in Huang et al. (Learning to predict arbitrary quantum processes, 2022. arXiv preprint arXiv:2210.14894). 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.
The research of A.V. is supported by NSF DMS-1900286, DMS-2154402 and by Hausdorff Center for Mathematics. H.Z. is supported by the Lise Meitner fellowship, Austrian Science Fund (FWF) M3337. This work is partially supported by NSF DMS-1929284 while both authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Harmonic Analysis and Convexity program.
Volberg A, Zhang H. Noncommutative Bohnenblust–Hille inequalities. Mathematische Annalen. 2023. doi:10.1007/s00208-023-02680-0
Volberg, A., & Zhang, H. (2023). Noncommutative Bohnenblust–Hille inequalities. Mathematische Annalen. Springer Nature. https://doi.org/10.1007/s00208-023-02680-0
Volberg, Alexander, and Haonan Zhang. “Noncommutative Bohnenblust–Hille Inequalities.” Mathematische Annalen. Springer Nature, 2023. https://doi.org/10.1007/s00208-023-02680-0.
A. Volberg and H. Zhang, “Noncommutative Bohnenblust–Hille inequalities,” Mathematische Annalen. Springer Nature, 2023.
Volberg A, Zhang H. 2023. Noncommutative Bohnenblust–Hille inequalities. Mathematische Annalen.
Volberg, Alexander, and Haonan Zhang. “Noncommutative Bohnenblust–Hille Inequalities.” Mathematische Annalen, Springer Nature, 2023, doi:10.1007/s00208-023-02680-0.