Quantum Lower Bounds by Sample-to-Query Lifting
arXiv:2308.01794 · doi:10.1137/24M1638616
Abstract
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 for quantum Gibbs sampling at inverse temperature , showing that the quantum Gibbs sampler by Gilyén, Su, Low, and Wiebe (STOC 2019) is optimal. 3. A new lower bound for the entanglement entropy problem with gap , 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.
Final version. 49 pages, 4 figures. [v4]: Corrected theorem numbering. [v3]: More explanations and references, minor corrections. [v2]: An error was found in [v1, Corollary 2.8] and it is corrected to [v2, Lemma 2.8], with consequent modifications in Lemma 2.16 and Lemma 2.19
References in corpus (34)
- Quantum algorithm for solving linear systems of equations
- Quantum principal component analysis
- Quantum fingerprinting
- Hamiltonian Simulation by Qubitization
- Optimal Hamiltonian Simulation by Quantum Signal Processing
- Simulating Hamiltonian dynamics with a truncated Taylor series
- Efficient quantum algorithms for simulating sparse Hamiltonians
- Quantum algorithm for systems of linear equations with exponentially improved dependence on precision
- Quantum Metropolis Sampling
- Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer
- Near-optimal ground state preparation
- Topological Order at Non-zero Temperature
- A Quantum-Quantum Metropolis Algorithm
- Quantum query complexity of some graph problems
- Feasibility of self-correcting quantum memory and thermal stability of topological order
- Quantum SDP-Solvers: Better upper and lower bounds
- Hamiltonian Simulation with Optimal Sample Complexity
- Quantum algorithm for estimating Renyi entropies of quantum states
- Quantum Algorithm for Fidelity Estimation
- Span-program-based quantum algorithm for evaluating formulas
- Quantum algorithms for testing properties of distributions
- Measuring Quantum Entropy
- Property testing of unitary operators
- Estimating distinguishability measures on quantum computers
- New Quantum Algorithms for Computing Quantum Entropies and Distances
- Fast Quantum Algorithms for Trace Distance Estimation
- A Lower Bound for Quantum Phase Estimation
- Quantum Time-Space Tradeoff for Finding Multiple Collision Pairs
- Optimal Trace Distance and Fidelity Estimations for Pure Quantum States
- Quantum algorithms for matrix geometric means
- A Quantum Algorithm Framework for Discrete Probability Distributions with Applications to Rényi Entropy Estimation
- Lower Bounds for Unitary Property Testing with Proofs and Advice
- Sample-based Hamiltonian and Lindbladian simulation: Non-asymptotic analysis of sample complexity
- Quantum state testing beyond the polarizing regime and quantum triangular discrimination