一个（几乎）高效的经典算法，用于从典型吉布斯态中采样
An (almost) efficient classical algorithm for sampling from typical Gibbs states
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中文总结 AI 辅助
本研究提出一种基于算法随机定位的经典采样算法，用于量子p-自旋模型的吉布斯态，证明在自旋玻璃转变温度以下，经典算法几乎高效，从而否定指数级量子优势。
中文摘要 AI 辅助
吉布斯态制备已成为物理系统研究中实现量子优势的潜在途径。尽管已知在计算最坏情况吉布斯态的局部性质时存在优势，但最近的结果表明，对于平均情况下的局部系统，情况有所不同：在温度足够低以至于其吉布斯态可以被量子高效制备的情况下，准多项式时间的经典算法也能计算局部期望值。然而，这仍然留下了在更强的模拟概念——采样——下存在优势的可能性。吉布斯采样是玻尔兹曼机为基础的量子学习算法和量子霸权提议的基础，但其在物理相关模型上的经典平均情况复杂度仍未解决。在此，我们提供了强有力的证据，反对在量子$p$-自旋模型这一局部系统系综的吉布斯采样中存在指数级量子优势。我们描述了一种经典算法，用于从吉布斯态采样，使得总变差距离误差达到任意多项式小。我们的算法基于一种称为算法随机定位（ASL）的元算法。我们的主要技术成果是ASL的广义收敛定理，该定理将超立方体上任意分布的近似采样问题简化为估计相关“倾斜分布”的均值，且加性误差足够低。然后，我们将已知的用于计算相关无序模型局部期望值的准多项式时间算法扩展，以计算量子$p$-自旋模型的这些“倾斜均值”。最后，我们给出了一个非严格的物理论证，表明该均值估计算法在自旋玻璃转变温度以下仍然有效。由于高效的量子算法在此转变点也会失效，这表明在温度低于经典算法（几乎）高效的温度时，量子算法无法对量子$p$-自旋模型进行吉布斯采样。
英文摘要
Gibbs state preparation has emerged as a potential avenue for quantum advantage in the study of physical systems. Although there is known advantage for computing local properties of worst-case Gibbs states, recent results suggest a different picture for average-case local systems: at temperatures where their Gibbs states can be efficiently prepared quantumly, quasi-polynomial time classical algorithms can also compute local expectation values. Yet this leaves open the possibility of an advantage under a stronger notion of simulation: sampling. Gibbs sampling underlies Boltzmann machine-based quantum learning algorithms and quantum supremacy proposals, but its classical average-case complexity for physically relevant models remains unresolved. Here, we give strong evidence against an exponential quantum advantage in Gibbs sampling for an ensemble of local systems known as the quantum $p$-spin model. We describe a classical algorithm for sampling from the Gibbs state to any polynomially small error in total variation distance. Our algorithm is based on a meta-algorithm known as algorithmic stochastic localization (ASL). Our main technical result is a generalized convergence theorem for ASL which reduces approximate sampling from any distribution on the hypercube to estimating the mean of a related "tilted distribution" to sufficiently low additive error. We then extend known quasi-polynomial time algorithms for computing local expectation values of a related disordered model to compute these "tilted means" for the quantum $p$-spin model. Finally, we give a non-rigorous physics argument that this mean-estimation algorithm works down to the spin glass transition temperature. As efficient quantum algorithms also fail at this transition, this suggests they cannot Gibbs sample the quantum $p$-spin model at temperatures below that for which classical algorithms are also (almost) efficient.
发表机构
- Institute for Quantum Information and Matter, Caltech（加州理工学院量子信息与物质研究所）
- Department of Physics, National University of Singapore（新加坡国立大学物理系）
- Department of Computer Science, National University of Singapore（新加坡国立大学计算机科学系）
- Walter Burke Institute for Theoretical Physics, Caltech（加州理工学院沃尔特·伯克理论物理研究所）
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