---
_id: '13318'
abstract:
- lang: eng
  text: 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.
acknowledgement: 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.
article_processing_charge: Yes (in subscription journal)
article_type: original
arxiv: 1
author:
- first_name: Alexander
  full_name: Volberg, Alexander
  last_name: Volberg
- first_name: Haonan
  full_name: Zhang, Haonan
  id: D8F41E38-9E66-11E9-A9E2-65C2E5697425
  last_name: Zhang
  orcid: 0000-0001-9537-9663
citation:
  ama: Volberg A, Zhang H. Noncommutative Bohnenblust–Hille inequalities. <i>Mathematische
    Annalen</i>. 2024;389:1657-1676. doi:<a href="https://doi.org/10.1007/s00208-023-02680-0">10.1007/s00208-023-02680-0</a>
  apa: Volberg, A., &#38; Zhang, H. (2024). Noncommutative Bohnenblust–Hille inequalities.
    <i>Mathematische Annalen</i>. Springer Nature. <a href="https://doi.org/10.1007/s00208-023-02680-0">https://doi.org/10.1007/s00208-023-02680-0</a>
  chicago: Volberg, Alexander, and Haonan Zhang. “Noncommutative Bohnenblust–Hille
    Inequalities.” <i>Mathematische Annalen</i>. Springer Nature, 2024. <a href="https://doi.org/10.1007/s00208-023-02680-0">https://doi.org/10.1007/s00208-023-02680-0</a>.
  ieee: A. Volberg and H. Zhang, “Noncommutative Bohnenblust–Hille inequalities,”
    <i>Mathematische Annalen</i>, vol. 389. Springer Nature, pp. 1657–1676, 2024.
  ista: Volberg A, Zhang H. 2024. Noncommutative Bohnenblust–Hille inequalities. Mathematische
    Annalen. 389, 1657–1676.
  mla: Volberg, Alexander, and Haonan Zhang. “Noncommutative Bohnenblust–Hille Inequalities.”
    <i>Mathematische Annalen</i>, vol. 389, Springer Nature, 2024, pp. 1657–76, doi:<a
    href="https://doi.org/10.1007/s00208-023-02680-0">10.1007/s00208-023-02680-0</a>.
  short: A. Volberg, H. Zhang, Mathematische Annalen 389 (2024) 1657–1676.
corr_author: '1'
das_tickbox: '1'
dataavailabilitystatement: All data generated or analysed during this study are included
  in this article.
date_created: 2023-07-30T22:01:03Z
date_published: 2024-06-01T00:00:00Z
date_updated: 2026-10-02T12:12:33Z
day: '01'
ddc:
- '510'
department:
- _id: JaMa
doi: 10.1007/s00208-023-02680-0
external_id:
  arxiv:
  - '2210.14468'
  isi:
  - '001035665500001'
  pmid:
  - '38751410'
file:
- access_level: open_access
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  creator: dernst
  date_created: 2024-07-22T09:38:15Z
  date_updated: 2024-07-22T09:38:15Z
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fulldoi: https://doi.org/10.1007/s00208-023-02680-0
has_accepted_license: '1'
intvolume: '       389'
isi: 1
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
page: 1657-1676
pmid: 1
project:
- _id: eb958bca-77a9-11ec-83b8-c565cb50d8d6
  grant_number: M03337
  name: Curvature-dimension in noncommutative analysis
publication: Mathematische Annalen
publication_identifier:
  eissn:
  - 1432-1807
  issn:
  - 0025-5831
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
researchdata_availability: yes
scopus_import: '1'
status: public
supplementarymaterial: no
title: Noncommutative Bohnenblust–Hille inequalities
tmp:
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  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 389
year: '2024'
...