Quantum Lower Bounds by Sample-to-Query Lifting

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.