Quantum hypothesis testing (QHT) has been traditionally studied from the information-theoretic perspective, wherein one is interested in the optimal decay rate of error probabilities as a function of the number of samples of an unknown state. In this paper, we study the sample complexity of QHT, wherein the goal is to determine the minimum number of samples needed to reach a desired error probability. By making use of the wealth of knowledge that already exists in the literature on QHT, we characterize the sample complexity of binary QHT in the symmetric and asymmetric settings, and we provide bounds on the sample complexity of multiple QHT. In more detail, we prove that the sample complexity of symmetric binary QHT depends logarithmically on the inverse error probability and inversely on the negative logarithm of the fidelity. As a counterpart of the quantum Stein's lemma, we also find that the sample complexity of asymmetric binary QHT depends logarithmically on the inverse type II error probability and inversely on the quantum relative entropy, provided that the type II error probability is sufficiently small. We then provide lower and upper bounds on the sample complexity of multiple QHT, with it remaining an intriguing open question to improve these bounds. The final part of our paper outlines and reviews how sample complexity of QHT is relevant to a broad swathe of research areas and can enhance understanding of many fundamental concepts, including quantum algorithms for simulation and search, quantum learning and classification, and foundations of quantum mechanics. As such, we view our paper as an invitation to researchers coming from different communities to study and contribute to the problem of sample complexity of QHT, and we outline a number of open directions for future research.

翻译：量子假设检验（QHT）传统上从信息论角度研究，关注未知态样本数变化时错误概率的最优衰减速率。本文研究QHT的样本复杂度，旨在确定达到目标错误概率所需的最小样本数。通过利用QHT文献中已有的丰富理论，我们刻画了对称与非对称二值QHT的样本复杂度，并给出了多值QHT样本复杂度的界限。具体而言，我们证明对称二值QHT的样本复杂度与逆错误概率的对数成正比，与保真度负对数成反比。作为量子Stein引理的对应结果，我们还发现当类型II错误概率充分小时，非对称二值QHT的样本复杂度与逆类型II错误概率的对数成正比，与量子相对熵成反比。随后，我们给出了多值QHT样本复杂度的下界与上界，改进这些界限仍是一个值得探索的开放问题。论文最后部分概述并评述了QHT样本复杂度如何与广泛的研究领域相关，并能加深对许多基本概念的理解，包括用于模拟和搜索的量子算法、量子学习与分类，以及量子力学基础。因此，我们将本文视为邀请不同领域研究者共同关注并推动QHT样本复杂度研究的指南，并为未来研究提出了若干开放性方向。