Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Quantum Learning Menagerie (A survey on Quantum learning for Classical concepts)

Sagnik Chatterjee

TL;DR

This survey analyzes learning classical concepts encoded in quantum states within the PAC framework, focusing on separations across query, sample, and time complexity between classical and quantum learners under various labeling oracles. It systematically catalogs oracle models (EX, MEM, QEX, QMEM, QREX, QAEX, MOS) and summarizes how quantum access affects learnability and complexity, including precise realizable and agnostic bounds such as $\Theta(d/\varepsilon + \log(1/\delta)/\varepsilon)$ in the realizable setting and $\Theta(d/\varepsilon^2 + \log(1/\delta)/\varepsilon^2)$ in the agnostic setting for quantum PAC sample complexity. The time-complexity discussions tie quantum speedups to structured problems like the Hidden Subgroup Problem (HSP) and lattice-based challenges (LWE/LPN), illustrating strong quantum advantages in certain regimes (e.g., Abelian HSP, dihedral cases) and challenging open questions in non-Abelian HSP, DNFs, decision trees, juntas, and shallow circuits. The work also clarifies the limits of current understanding by highlighting 23 open problems and exploring generalized oracle models (MOS) and white-box/unitary-access scenarios, with implications for cryptography and quantum learning theory. Overall, the paper delineates when quantum data access yields true computational benefits for learning classical concepts and maps out promising directions for future research.

Abstract

This paper surveys various results in the field of Quantum Learning theory, specifically focusing on learning quantum-encoded classical concepts in the Probably Approximately Correct (PAC) framework. The cornerstone of this work is the emphasis on query, sample, and time complexity separations between classical and quantum learning that emerge under learning with query access to different labeling oracles. This paper aims to consolidate all known results in the area under the above umbrella and underscore the limits of our understanding by leaving the reader with 23 open problems.

The Quantum Learning Menagerie (A survey on Quantum learning for Classical concepts)

TL;DR

This survey analyzes learning classical concepts encoded in quantum states within the PAC framework, focusing on separations across query, sample, and time complexity between classical and quantum learners under various labeling oracles. It systematically catalogs oracle models (EX, MEM, QEX, QMEM, QREX, QAEX, MOS) and summarizes how quantum access affects learnability and complexity, including precise realizable and agnostic bounds such as in the realizable setting and in the agnostic setting for quantum PAC sample complexity. The time-complexity discussions tie quantum speedups to structured problems like the Hidden Subgroup Problem (HSP) and lattice-based challenges (LWE/LPN), illustrating strong quantum advantages in certain regimes (e.g., Abelian HSP, dihedral cases) and challenging open questions in non-Abelian HSP, DNFs, decision trees, juntas, and shallow circuits. The work also clarifies the limits of current understanding by highlighting 23 open problems and exploring generalized oracle models (MOS) and white-box/unitary-access scenarios, with implications for cryptography and quantum learning theory. Overall, the paper delineates when quantum data access yields true computational benefits for learning classical concepts and maps out promising directions for future research.

Abstract

This paper surveys various results in the field of Quantum Learning theory, specifically focusing on learning quantum-encoded classical concepts in the Probably Approximately Correct (PAC) framework. The cornerstone of this work is the emphasis on query, sample, and time complexity separations between classical and quantum learning that emerge under learning with query access to different labeling oracles. This paper aims to consolidate all known results in the area under the above umbrella and underscore the limits of our understanding by leaving the reader with 23 open problems.
Paper Structure (54 sections, 16 theorems, 31 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 54 sections, 16 theorems, 31 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Structure and Organisation
  3. Background and Preliminaries
  4. Basic Notation
  5. Basics of Quantum Computing
  6. Overview of Computational Learning Theory
  7. The PAC-learning framework
  8. Sample Complexity and Learnability
  9. Time Complexity and Efficient Learnability
  10. Query Complexity Separations
  11. PAC Learning with queries
  12. PAC learning with Sampling Oracles
  13. Classical Example Query Oracles
  14. Quantum Example Query Oracles
  15. PAC learning with Membership Query Oracles
  16. ...and 39 more sections

Key Result

Lemma 3.1

Any $n$-bit Boolean function is PAC learnable with Membership queries in polynomial time in $n$ and its DNF size or its decision tree size.

Figures (1)

  • Figure 1: Some common oracles used in quantum algorithms

Theorems & Definitions (63)

  • Definition 2.1: Trace Distance
  • Remark 2.1
  • Remark 2.2
  • Definition 2.2: Error
  • Remark 2.3
  • Definition 2.3: $(\varepsilon,\delta)$-PAC learnability
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Definition 3.1: PAC learning with Oracle access
  • ...and 53 more