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.

翻译：尽管在某些情境下，利用量子样本学习概念类可能比经典样本更高效，但Arunachalam与de Wolf（JMLR, 2018）证明，在量子PAC与不可知学习模型中，量子学习器渐近意义上并不比经典学习器更高效。他们通过量子态识别与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，在PAC与不可知模型中推导了量子样本复杂度的最优下界。其证明过程更为简洁，且相同思路或可推广至量子学习理论其他问题的最优界推导。随后我们转向量子“优惠券收集者”问题——这一概率论经典问题在PAC学习研究中亦具重要性。Arunachalam、Belovs、Childs、Kothari、Rosmanis与de Wolf（TQC, 2020）已将该问题的量子样本复杂度刻画至常数因子。首先，我们证明上述信息论方法无法给出最优下界。作为副产品，我们构造了一个任意高维下的自然纯态系综，该系综难以被（同时）区分，但其Holevo信息接近最大值。其次，我们发现信息论方法能为该问题的近似变体提供渐近最优界。最后，通过广义Holevo-Curlander系综可区分性界，我们推导出量子优惠券收集者问题更紧的下界。我们所研究的量子优惠券收集者问题的所有方面，均依赖于关联Gram矩阵谱的性质，该性质本身可能具有独立的研究价值。