🤖 AI Summary
This work addresses the lack of systematic tools for proving quantum query lower bounds. We establish the first quantum sample-to-query lifting theorem, which rigorously connects sampling complexity to query complexity from an information-theoretic perspective—introducing a new paradigm for quantum lower-bound analysis. The theorem unifies analyses across several fundamental problems: quantum property testing, Gibbs sampling, entanglement entropy estimation, and matrix spectral testing. Specifically, it yields the optimal quadratic separation for quantum state discrimination; proves the query-optimality of Gilyén et al.’s Gibbs sampler; provides a tight (widetilde{Omega}(1/sqrt{Delta})) lower bound on entanglement entropy estimation under (Delta)-spectral gap; and strengthens existing lower bounds for multiple spectral testing tasks. The result combines deep theoretical insight with broad methodological applicability, offering a unified framework for deriving quantum query lower bounds across diverse computational settings.
📝 Abstract
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 $widetilde Omega(eta)$ for quantum Gibbs sampling at inverse temperature $eta$, 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.
