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Equivalence of Privacy and Stability with Generalization Guarantees in Quantum Learning

Ayanava Dasgupta, Naqueeb Ahmad Warsi, Masahito Hayashi

TL;DR

The paper develops an information-theoretic framework connecting quantum differential privacy, algorithmic stability, and generalization, showing that a $1$-neighbor $(\varepsilon,\delta)$-QDP constraint yields a mutual-information bound that governs generalization in quantum learners. By introducing Classical-Quantum Sub-Gaussianity, it derives a mechanism-agnostic generalization bound that scales with the MI term $I[S\bm{\mathfrak{Te}};WB']$, and it extends the analysis to untrusted data processors via Information-Theoretic Admissibility (ITA). A grid-covering technique provides a Holevo-information bound that recovers classical results as a special case and demonstrates that privacy guarantees can persist under ITA in the quantum setting due to non-commutativity. The results offer a principled, quantum-specific pathway to privacy-preserving learning with provable generalization, and they reveal a quantum advantage over classical privacy notions when dealing with adversarial processors. These insights have potential implications for designing privacy-aware quantum learning systems and for understanding the fundamental limits of private quantum information processing.

Abstract

We present a unified information-theoretic framework to analyze the generalization performance of differentially private (DP) quantum learning algorithms. By leveraging the connection between privacy and algorithmic stability, we establish that $(\varepsilon, δ)$-Quantum Differential Privacy (QDP) imposes a strong constraint on the mutual information between the training data and the algorithm's output. We derive a rigorous, mechanism-agnostic upper bound on this mutual information for learning algorithms satisfying a 1-neighbor privacy constraint. Furthermore, we connect this stability guarantee to generalization, proving that the expected generalization error of any $(\varepsilon, δ)$-QDP learning algorithm is bounded by the square root of the privacy-induced stability term. Finally, we extend our framework to the setting of an untrusted Data Processor, introducing the concept of Information-Theoretic Admissibility (ITA) to characterize the fundamental limits of privacy in scenarios where the learning map itself must remain oblivious to the specific dataset instance.

Equivalence of Privacy and Stability with Generalization Guarantees in Quantum Learning

TL;DR

The paper develops an information-theoretic framework connecting quantum differential privacy, algorithmic stability, and generalization, showing that a -neighbor -QDP constraint yields a mutual-information bound that governs generalization in quantum learners. By introducing Classical-Quantum Sub-Gaussianity, it derives a mechanism-agnostic generalization bound that scales with the MI term , and it extends the analysis to untrusted data processors via Information-Theoretic Admissibility (ITA). A grid-covering technique provides a Holevo-information bound that recovers classical results as a special case and demonstrates that privacy guarantees can persist under ITA in the quantum setting due to non-commutativity. The results offer a principled, quantum-specific pathway to privacy-preserving learning with provable generalization, and they reveal a quantum advantage over classical privacy notions when dealing with adversarial processors. These insights have potential implications for designing privacy-aware quantum learning systems and for understanding the fundamental limits of private quantum information processing.

Abstract

We present a unified information-theoretic framework to analyze the generalization performance of differentially private (DP) quantum learning algorithms. By leveraging the connection between privacy and algorithmic stability, we establish that -Quantum Differential Privacy (QDP) imposes a strong constraint on the mutual information between the training data and the algorithm's output. We derive a rigorous, mechanism-agnostic upper bound on this mutual information for learning algorithms satisfying a 1-neighbor privacy constraint. Furthermore, we connect this stability guarantee to generalization, proving that the expected generalization error of any -QDP learning algorithm is bounded by the square root of the privacy-induced stability term. Finally, we extend our framework to the setting of an untrusted Data Processor, introducing the concept of Information-Theoretic Admissibility (ITA) to characterize the fundamental limits of privacy in scenarios where the learning map itself must remain oblivious to the specific dataset instance.
Paper Structure (37 sections, 8 theorems, 57 equations, 3 figures)

This paper contains 37 sections, 8 theorems, 57 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background and Motivation (Classical to Quantum)
  3. Our Framework and Main Contributions
  4. Organization of this Manuscript
  5. Notations
  6. Method of Types.
  7. A General Framework for Quantum Learning Algorithms
  8. Learning Setup and Data Encoding
  9. Stability of a Quantum Learning Algorithm
  10. Stability Implies Generalizability For Quantum Learning Algorithms
  11. Generalization Guarantees for Differentially Private Quantum Learning
  12. Trusted Setting and One-Neighbor $(\varepsilon,\delta)$-DP
  13. Privacy Implies Stability: A Mutual-Information Bound
  14. From Stability to Generalization: DP Generalization Guarantees
  15. Untrusted Data Processor and Information Theoretic Admissibility (ITA)
  16. ...and 22 more sections

Key Result

Theorem 1

For a fixed $\alpha \in (0,\infty),$ if the loss operators for a quantum learning algorithm $\mathcal{N}$, satisfy Definition ass_cq_subg, then, we have,

Figures (3)

  • Figure 1: Privacy based learning framework.
  • Figure 2: Privacy based learning framework.
  • Figure 3: Numerical comparison of the generalization error bounds for the classical-quantum toy example in WDH2025. (a) Comparison of $\mathcal{B}_{\text{MI}}$\ref{['B_IT']} and $\mathcal{B}_{\text{SEP}}$\ref{['B_SEP']} as a function of the prior probability $p \in [0.25, 0.75]$. (b) Comparison of $\mathcal{B}_{\text{MI}}$\ref{['B_IT']} and $\mathcal{B}_{\text{SEP}}$\ref{['B_SEP']} as a function of the sub-Gaussianity parameter $\alpha \in [0.1, 1]$ for a fixed prior $p=0.4$. In both regimes, our bound $\mathcal{B}_{\text{MI}}$ (blue) provides a strictly tighter upper bound than $\mathcal{B}_{\text{SEP}}$ (orange).

Theorems & Definitions (31)

  • Definition 1
  • Definition 2: Expected Empirical Loss Caro23
  • Definition 3: Expected True Loss WDH2025
  • Remark 1
  • Definition 4: Expected Generalization Error WDH2025
  • Definition 5: Classical-Quantum Sub-Gaussianity
  • Theorem 1
  • proof
  • Remark 2
  • Corollary 1
  • ...and 21 more