发表机构
University of California, Los Angeles; Google Quantum AI; Harvard University（加州大学洛杉矶分校; 谷歌量子人工智能; 哈佛大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究将吉布斯采样归约为量子解码，证明DQI可突破碎裂相中的拓扑障碍，在超越动力学相变阈值的温度下实现经典伊辛自旋玻璃的高效采样。
AI 中文摘要
我们将解码量子干涉测量（DQI）应用于经典伊辛自旋哈密顿量的吉布斯测度采样。我们证明，这一吉布斯采样问题可归结为量子解码问题，且DQI可达到的温度由解码算法的性能决定。随后，我们聚焦于平均度$D\ge k$的随机埃尔德什-雷尼超图上经典伊辛$k$-自旋玻璃（或Max-$k$-XORSAT）的吉布斯采样任务。在渐近起始于预测动力学相变$\beta_{\rm dyn}(k,D) = \sqrt{(2\ln k)/D}\times [1+o_{k\to\infty}(1)]$的温度范围内，我们证明碎裂与无序混沌构成拓扑障碍，阻碍许多算法，包括格劳伯动力学以及任何输出分布在输入扰动下“稳定”的算法。相反，我们证明这一障碍可被基于普朗奇方法的经典算法以及配备量子解码器的DQI所打破。例如，当$D=\alpha k$且固定$\alpha>1$时，对于足够大的$k$，普朗奇算法和DQI均可在任意逆温度$\beta < \tanh^{-1}(1/\alpha)$下采样，远超动力学阈值$\beta_{\rm dyn} \sim \sqrt{2\ln k / (\alpha k)}$。因此，我们的结果表明，DQI能够克服阻碍稳定算法的拓扑障碍。
英文摘要
We apply Decoded Quantum Interferometry (DQI) to sample from the Gibbs measures of classical Ising spin Hamiltonians. We show that this Gibbs sampling problem reduces to a quantum decoding problem, and the temperature achievable by DQI is determined by the performance of decoding algorithms. We then focus on the task of Gibbs sampling for classical Ising $k$-spin glasses (or Max-$k$-XORSAT) on random Erdős-Rényi hypergraphs with average degree $D\ge k$. In a temperature range beginning asymptotically at the predicted dynamical phase transition, $β_{\rm dyn}(k,D) = \sqrt{(2\ln k)/D}\times [1+o_{k\to\infty}(1)]$, we show that shattering and disorder chaos form a topological barrier that obstructs many algorithms, including Glauber dynamics and any algorithm whose output distribution is "stable" under perturbations of the input. In contrast, we prove that this barrier can be broken both by a classical algorithm based on Prange's method, and by DQI equipped with a quantum decoder. For example, when $D=αk$ with fixed $α>1$, both Prange's algorithm and DQI can sample at any inverse temperature $β< \tanh^{-1}(1/α)$ for sufficiently large $k$, well beyond the dynamical threshold $β_{\rm dyn} \sim \sqrt{2\ln k / (αk)}$. Therefore, our results show that DQI can overcome topological barriers that obstruct stable algorithms.
Comments44 pages, 1 figure, 3 tables
