  The polynomial method by Beals, Buhrman, Cleve, Mosca, and de Wolf (FOCS
1998) and the adversary method by Ambainis (STOC 2000) have been shown to be
powerful in proving quantum query lower bounds for a wide variety of problems.
In this paper, we propose an arguably 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.
  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).
  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.
