连续Gibbs采样的可证量子-经典分离
Provable Quantum-Classical Separation for Continuous Gibbs Sampling
- Irréversible Inc.（不可逆公司）
- Institute for Quantum Computing, University of Waterloo（滑铁卢大学量子计算研究所）
- Department of Physics and Astronomy, University of Waterloo（滑铁卢大学物理与天文学系）
- Perimeter Institute for Theoretical Physics（圆周理论物理研究所）
- National Research Council Canada（加拿大国家研究委员会）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究证明了连续域Gibbs采样问题的首个量子-经典分离，经典算法需Ω(α)次查询，而基于量子奇异值阈值化和温度退火的量子算法仅需Õ(√α)次查询，优势在低温下随维度指数增长。
AI中文摘要：
我们证明了首个连续域采样问题的量子-经典分离。对于环面$\u2164^d$上一类Gibbs态$p\propto e^{-βE}$，其势函数为光滑（$s$-Gevrey）势，势垒幅度$α=e^{βΔ}$（其中$Δ= \max E-\min E$），所有经典算法——可查询对数密度的取值、梯度或任意高阶导数——都需要$Ω(α)$次查询才能在总变分距离下以恒定精度采样，而基于量子奇异值阈值化和温度退火的量子算法，仅需对梯度预言机进行$\tilde{O}\left(\sqrtα\right)$次查询即可完成采样。该优势在势垒幅度上呈二次方关系，在低温下随维度呈指数增长，达到$e^{Ω(d)}$。该经典下界是信息论意义上的，适用于所有可查询Gibbs势及其任意阶导数的经典算法。
英文摘要：
We prove the first quantum-classical separation for a sampling problem over a continuous domain. For a class of Gibbs states $p\propto e^{-βE}$ on the torus $\mathbb{T}^d$ with smooth ($s$-Gevrey) potential and barrier amplitude $α=e^{βΔ}$, where $Δ= \max E-\min E$, every classical algorithm querying the value, gradient, or any higher-order derivatives of the log-density requires $Ω(α)$ queries to sample at constant accuracy in total variation distance, while a quantum algorithm based on quantum singular value thresholding and temperature annealing samples with $\tilde{O}\left(\sqrtα\right)$ queries to an oracle for the gradient. The advantage is quadratic in the barrier amplitude, which becomes exponential in the dimension, $e^{Ω(d)}$, at low temperature. The classical bound is information-theoretic, holding for every classical algorithm with query access to the Gibbs potential and its derivatives at any order.
