We propose a quantum sample-to-query lifting theorem. It reveals a quadratic relation between quantum sample and query complexities regarding quantum property testing, which is optimal and saturated by quantum state discrimination. Based on it, we provide a new method for proving lower bounds on quantum query algorithms from an information theory perspective. Using this method, we prove the following new results: 1. 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 (2019) is optimal. 2. 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 (2023). 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：我们提出一个量子样本到查询的提升定理。该定理揭示了量子属性测试中样本复杂度与查询复杂度之间的二次关系，这一关系在量子态区分问题中达到最优且饱和。基于此，我们提出一种从信息论视角证明量子查询算法下界的新方法。利用该方法，我们证明了以下新结果：1. 逆温度β下量子吉布斯采样的匹配下界$\widetilde \Omega(\beta)$，表明Gily\'en、Su、Low和Wiebe（2019）的量子吉布斯采样器是最优的；2. 能隙为Δ的纠缠熵问题的新下界$\widetilde \Omega(1/\sqrt{\Delta})$，该问题最近由She和Yuen（2023）研究。此外，我们还为一些已知下界提供了统一证明，这些下界此前通过不同技术得到证明，包括相位/振幅估计和哈密顿量模拟中的下界。