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 sharp lower bound for the Quantum Coupon Collector problem, with the exact leading order term, 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.

翻译：尽管在某些场景下，使用量子样本相较于经典样本学习概念类可能更高效，但Arunachalam与de Wolf（JMLR, 2018）证明，在量子PAC和不可知学习模型中，量子学习器的渐近效率并不优于经典学习器。他们通过量子态识别和傅里叶分析建立了样本复杂度的下界。本文采用信息论方法，推导了PAC和不可知模型下量子样本复杂度的最优下界。其证明过程更为简洁，且相同思想或可用于推导量子学习理论中其他问题的最优下界。随后，我们转而研究量子版的“优惠券收集”问题——该经典概率论问题在PAC学习研究中也具有重要性。Arunachalam、Belovs、Childs、Kothari、Rosmanis与de Wolf（TQC, 2020）刻画了该问题的量子样本复杂度（常系数精度内）。首先，我们证明上述信息论方法无法得出最优下界。作为副产品，我们构建了一个高维纯态的自然系综，该系综中的态难以（同时）区分，但其Holevo信息却接近最大值。其次，我们发现信息论方法能为该问题的近似变体给出渐近最优界。最后，通过推广的Holevo-Curlander系综可分性界，我们推导了量子优惠券收集问题的尖锐下界（含精确首阶项）。我们研究的所有量子优惠券收集问题方面均依赖于关联Gram矩阵谱的性质，该性质本身可能具有独立研究价值。