Naqueeb Ahmad Warsi

Naqueeb Ahmad Warsi, Ayanava Dasgupta, Masahito Hayashi

This work advances the theoretical understanding of quantum learning by establishing a new family of upper bounds on the expected generalization error of quantum learning algorithms, leveraging the framework introduced by Caro et al. (2024) and a new definition for the expected true loss. Our primary contribution is the derivation of these bounds in terms of quantum and classical Rényi divergences, utilizing a variational approach for evaluating quantum Rényi divergences, specifically the Petz and a newly introduced modified sandwich quantum Rényi divergence. Analytically and numerically, we demonstrate the superior performance of the bounds derived using the modified sandwich quantum Rényi divergence compared to those based on the Petz divergence. Furthermore, we provide probabilistic generalization error bounds using two distinct techniques: one based on the modified sandwich quantum Rényi divergence and classical Rényi divergence, and another employing smooth max Rényi divergence.

Ayanava Dasgupta, Naqueeb Ahmad Warsi, Masahito Hayashi

We study quantum differential privacy (QDP) by defining a notion of the order of informativeness between two pairs of quantum states. In particular, we show that if the hypothesis testing divergence of the one pair dominates over that of the other pair, then this dominance holds for every $f$-divergence. This approach completely characterizes $(\varepsilon,δ)$-QDP mechanisms by identifying the most informative $(\varepsilon,δ)$-DP quantum state pairs. We apply this to analyze the stability of quantum differentially private learning algorithms, generalizing classical results to the case $δ>0$. Additionally, we study precise limits for privatized hypothesis testing and privatized quantum parameter estimation, including tight upper-bounds on the quantum Fisher information under QDP. Finally, we establish near-optimal contraction bounds for differentially private quantum channels with respect to the hockey-stick divergence.

Ayanava Dasgupta, Naqueeb Ahmad Warsi, Masahito Hayashi

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.