Quantum and Classical Low-Degree Learning via a Dimension-Free Remez Inequality

Ohad Klein, Joseph Slote, Alexander Volberg, Haonan Zhang

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Recent efforts in Analysis of Boolean Functions aim to extend core results to new spaces, including to the slice [n] k , the hypergrid [K]n, and noncommutative spaces (matrix algebras). We present here a new way to relate functions on the hypergrid (or products of cyclic groups) to their harmonic extensions over the polytorus. We show the supremum of a function f over products of the cyclic group {exp(2 ik/K)}Kk =1 controls the supremum of f over the entire polytorus ({z C : z = 1}n), with multiplicative constant C depending on K and deg(f) only. This Remez-Type inequality appears to be the first such estimate that is dimension-free (i.e., C does not depend on n). This dimension-free Remez-Type inequality removes the main technical barrier to giving O(log n) sample complexity, polytime algorithms for learning low-degree polynomials on the hypergrid and low-degree observables on level-K qudit systems. In particular, our dimension-free Remez inequality implies new Bohnenblust-Hille-Type estimates which are central to the learning algorithms and appear unobtainable via standard techniques. Thus we extend to new spaces a recent line of work [10, 13, 23] that gave similarly efficient methods for learning low-degree polynomials on the hypercube and observables on qubits. An additional product of these efforts is a new class of distributions over which arbitrary quantum observables are well-Approximated by their low-degree truncations - a phenomenon that greatly extends the reach of low-degree learning in quantum science [13].

Original languageEnglish
Title of host publication15th Innovations in Theoretical Computer Science Conference, ITCS 2024
EditorsVenkatesan Guruswami
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959773096
DOIs
StatePublished - Jan 2024
Externally publishedYes
Event15th Innovations in Theoretical Computer Science Conference, ITCS 2024 - Berkeley, United States
Duration: 30 Jan 20242 Feb 2024

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume287
ISSN (Print)1868-8969

Conference

Conference15th Innovations in Theoretical Computer Science Conference, ITCS 2024
Country/TerritoryUnited States
CityBerkeley
Period30/01/242/02/24

Bibliographical note

Publisher Copyright:
© 2024 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.

Keywords

  • Analysis of Boolean Functions
  • Bohnenblust-Hille Inequality
  • Qudits
  • Remez Inequality
  • Statistical Learning Theory

Fingerprint

Dive into the research topics of 'Quantum and Classical Low-Degree Learning via a Dimension-Free Remez Inequality'. Together they form a unique fingerprint.

Cite this