The polynomial method by Beals, Buhrman, Cleve, Mosca, and de Wolf (FOCS 1998, J. ACM 2001), the adversary method by Ambainis (STOC 2000, J. Comput. Syst. Sci. 2002), and the compressed oracle method by Zhandry (CRYPTO 2019) have been shown to be powerful in proving quantum query lower bounds for a wide variety of problems. In this paper, we propose a new method for proving quantum query lower bounds by a quantum sample-to-query lifting theorem, which is from an information theory perspective. Using this method, we obtain the following new results: 1. A quadratic relation between quantum sample and query complexities regarding quantum property testing, which is optimal and saturated by quantum state discrimination. Here, the sample complexity is measured given sample access to the quantum state to be tested, while the query complexity is measured given query access to an oracle that block-encodes the quantum state. 2. A matching lower bound $\widetilde \Omega(\beta)$ for quantum Gibbs sampling at inverse temperature $\beta$, showing that the quantum Gibbs sampler by Gily\'en, Su, Low, and Wiebe (STOC 2019) is optimal. 3. A new lower bound $\widetilde \Omega(1/\sqrt{\Delta})$ for the entanglement entropy problem with gap $\Delta$, which was recently studied by She and Yuen (ITCS 2023). 4. A series of quantum query lower bounds for matrix spectrum testing, based on the sample lower bounds for quantum state spectrum testing by O'Donnell and Wright (STOC 2015, Comm. Math. Phys. 2021). In addition, we also provide unified proofs for some known lower bounds that have been proven previously via different techniques, including those for phase/amplitude estimation and Hamiltonian simulation.

翻译：Beals、Buhrman、Cleve、Mosca和de Wolf的多项式方法（FOCS 1998, J. ACM 2001）、Ambainis的对手方法（STOC 2000, J. Comput. Syst. Sci. 2002）以及Zhandry的压缩预言机方法（CRYPTO 2019）已被证明在证明各类问题的量子查询下界方面非常强大。本文提出一种基于量子样本-查询提升定理证明量子查询下界的新方法，该方法从信息论视角出发。利用此方法，我们获得了以下新结果：1. 关于量子性质测试的量子样本复杂度与查询复杂度之间的二次关系，该关系是最优的且被量子态区分问题所饱和。其中，样本复杂度是在给定对被测量子态的样本访问下度量的，而查询复杂度是在给定对块编码该量子态的预言机的查询访问下度量的。2. 逆温度 $\beta$ 下量子吉布斯采样的匹配下界 $\widetilde \Omega(\beta)$，表明Gilyén、Su、Low和Wiebe的量子吉布斯采样器（STOC 2019）是最优的。3. 对于间隙为 $\Delta$ 的纠缠熵问题的新下界 $\widetilde \Omega(1/\sqrt{\Delta})$，该问题最近由She和Yuen研究（ITCS 2023）。4. 基于O'Donnell和Wright关于量子态谱测试的样本下界（STOC 2015, Comm. Math. Phys. 2021），得到矩阵谱测试的一系列量子查询下界。此外，我们还为一些先前通过不同技术已证明的已知下界提供了统一证明，包括相位/幅度估计和哈密顿量模拟的下界。