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. 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开展研究，关注未知量子态样本数增加时错误概率的最优衰减率。本文研究QHT的样本复杂度问题，旨在确定达到期望错误概率所需的最小样本数。通过充分利用现有QHT文献中的丰富知识，我们刻画了对称与非对称设定下二元QHT的样本复杂度，并给出了多重QHT样本复杂度的上下界。具体而言，我们证明对称二元QHT的样本复杂度与错误概率倒数的对数成正比，与保真度负对数成反比。作为量子Stein引理的对偶结论，我们发现非对称二元QHT的样本复杂度与第二类错误概率倒数的对数成正比，与量子相对熵成反比。进一步地，我们给出了多重QHT样本复杂度的上下界，改进这些界限仍是引人入胜的开放问题。论文最后部分概述并系统阐释了QHT样本复杂度如何广泛关联多个研究领域，能增进对诸多基础概念的理解，包括量子模拟与搜索算法、量子学习与分类，以及量子力学基础。基于此，我们将本文视为向不同领域研究者发出的邀请，期待大家共同研究与贡献于QHT样本复杂度问题，并提出了若干未来可探索的开放研究方向。