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Optimal lower bounds for Quantum Learning via Information Theory

Shima Bab Hadiashar, Ashwin Nayak, Pulkit Sinha

TL;DR

The paper establishes optimal lower bounds on quantum sample complexity for PAC and agnostic learning by an information-theoretic analysis that directly bounds mutual information via the spectrum of the quantum sample state. It shows the standard subadditivity approach can be suboptimal and supplies a spectral Gram-matrix framework to obtain tight bounds, with tools from the Johnson association scheme and random-walk interpretations aiding the analysis. In studying the Quantum Coupon Collector, it demonstrates the limits of the naive information-theoretic route, yet achieves asymptotically optimal bounds for approximation and a sharp leading-term bound for the exact problem using generalized Holevo-Curlander bounds. Overall, the work provides a simpler, information-theoretic route to optimal quantum learning bounds and highlights spectral methods as broadly applicable in quantum learning theory.

Abstract

Although a concept class may be learnt more efficiently using quantum samples as compared with classical samples in certain scenarios, Arunachalam and de Wolf (JMLR, 2018) proved that quantum learners are asymptotically no more efficient than classical ones in the quantum PAC and Agnostic learning models. They established lower bounds on sample complexity via quantum state identification and Fourier analysis. In this paper, we derive optimal lower bounds for quantum sample complexity in both the PAC and agnostic models via an information-theoretic approach. The proofs are arguably simpler, and the same ideas can potentially be used to derive optimal bounds for other problems in quantum learning theory. We then turn to a quantum analogue of the Coupon Collector problem, a classic problem from probability theory also of importance in the study of PAC learning. Arunachalam, Belovs, Childs, Kothari, Rosmanis, and de Wolf (TQC, 2020) characterized the quantum sample complexity of this problem up to constant factors. First, we show that the information-theoretic approach mentioned above provably does not yield the optimal lower bound. As a by-product, we get a natural ensemble of pure states in arbitrarily high dimensions which are not easily (simultaneously) distinguishable, while the ensemble has close to maximal Holevo information. Second, we discover that the information-theoretic approach yields an asymptotically optimal bound for an approximation variant of the problem. Finally, we derive a sharper lower bound for the Quantum Coupon Collector problem, via the generalized Holevo-Curlander bounds on the distinguishability of an ensemble. All the aspects of the Quantum Coupon Collector problem we study rest on properties of the spectrum of the associated Gram matrix, which may be of independent interest.

Optimal lower bounds for Quantum Learning via Information Theory

TL;DR

The paper establishes optimal lower bounds on quantum sample complexity for PAC and agnostic learning by an information-theoretic analysis that directly bounds mutual information via the spectrum of the quantum sample state. It shows the standard subadditivity approach can be suboptimal and supplies a spectral Gram-matrix framework to obtain tight bounds, with tools from the Johnson association scheme and random-walk interpretations aiding the analysis. In studying the Quantum Coupon Collector, it demonstrates the limits of the naive information-theoretic route, yet achieves asymptotically optimal bounds for approximation and a sharp leading-term bound for the exact problem using generalized Holevo-Curlander bounds. Overall, the work provides a simpler, information-theoretic route to optimal quantum learning bounds and highlights spectral methods as broadly applicable in quantum learning theory.

Abstract

Although a concept class may be learnt more efficiently using quantum samples as compared with classical samples in certain scenarios, Arunachalam and de Wolf (JMLR, 2018) proved that quantum learners are asymptotically no more efficient than classical ones in the quantum PAC and Agnostic learning models. They established lower bounds on sample complexity via quantum state identification and Fourier analysis. In this paper, we derive optimal lower bounds for quantum sample complexity in both the PAC and agnostic models via an information-theoretic approach. The proofs are arguably simpler, and the same ideas can potentially be used to derive optimal bounds for other problems in quantum learning theory. We then turn to a quantum analogue of the Coupon Collector problem, a classic problem from probability theory also of importance in the study of PAC learning. Arunachalam, Belovs, Childs, Kothari, Rosmanis, and de Wolf (TQC, 2020) characterized the quantum sample complexity of this problem up to constant factors. First, we show that the information-theoretic approach mentioned above provably does not yield the optimal lower bound. As a by-product, we get a natural ensemble of pure states in arbitrarily high dimensions which are not easily (simultaneously) distinguishable, while the ensemble has close to maximal Holevo information. Second, we discover that the information-theoretic approach yields an asymptotically optimal bound for an approximation variant of the problem. Finally, we derive a sharper lower bound for the Quantum Coupon Collector problem, via the generalized Holevo-Curlander bounds on the distinguishability of an ensemble. All the aspects of the Quantum Coupon Collector problem we study rest on properties of the spectrum of the associated Gram matrix, which may be of independent interest.
Paper Structure (24 sections, 32 theorems, 144 equations)

This paper contains 24 sections, 32 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper.
  3. Acknowledgements.
  4. Preliminaries
  5. Some notation.
  6. Classical information theory.
  7. Tail bounds.
  8. Quantum Information Theory.
  9. Quantum learning theory.
  10. PAC model.
  11. Agnostic model.
  12. Quantum Coupon Collection.
  13. Learning Boolean functions
  14. Sample complexity of PAC learning
  15. Sample complexity of agnostic learning
  16. ...and 9 more sections

Key Result

Lemma 3.1

Let ${\mathscr C}$ be a non-trivial concept class. For every $\delta \in (0,1/2)$ and $\epsilon \in (0,1/4)$, an $(\epsilon, \delta)$-PAC quantum learner for ${\mathscr C}$ has quantum sample complexity at least $\Omega\mleft(\tfrac{1}{\epsilon} \log \tfrac{1}{\delta}\mright)$.

Theorems & Definitions (35)

  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1
  • Theorem 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Theorem 3.7
  • Lemma 3.8
  • ...and 25 more