用无监督学习发现三维与量子相变，精度接近理论值。
From Classical to Quantum: Extending Prometheus for Unsupervised Discovery of Phase Transitions in Three Dimensions and Quantum Systems
- 扩展框架至三维与量子系统，基于变分自编码器处理复数波函数
- 3D伊辛模型临界温度误差小于0.01%，量子临界点检测误差仅2%
- 可识别不同临界行为类型，适合物理与机器学习交叉研究者
我们将Prometheus框架从二维经典系统拓展至三维经典系统和量子多体系统。基于对二维伊辛模型的初步观察，解决两个核心问题：（1）该框架能否在无精确解的高维系统中有效？（2）能否泛化至由量子涨落驱动而非热涨落的量子相变？对于尺寸达$ L=32 $的三维伊辛模型，临界温度检测精度达文献值的0.01%以内（$ T_c/J = 4.511 \pm 0.005 $），临界指数提取准确率超过70%，统计分析正确识别3D伊辛普适类（$ p=0.72 $）。针对量子系统，我们构建量子感知变分自编码器（Q-VAE），在复数波函数上使用保真度损失函数，使横向场伊辛模型的量子临界点检测误差为2%。对于无序横向场伊辛模型，验证了激活动力学标度 $\ln ξ\sim |h - h_c|^{-ψ}$，提取隧道指数 $ψ= 0.48 \pm 0.08 $，与理论预测（$ψ= 0.5$）一致（$Δχ^2 = 12.3$，$p < 0.001$）。结果表明，无监督学习可识别不同类型临界行为，为已知红外固定点物理提供一致性检验。
原文摘要 · Abstract (English)
We extend the Prometheus framework for unsupervised phase transition discovery from two-dimensional classical systems to three-dimensional classical systems and quantum many-body systems. Building upon preliminary observations from a 2D Ising model student abstract [Yee et al., 2026], we address two fundamental questions: (1) Does the framework scale to higher dimensions where exact solutions are unavailable? (2) Can it generalize to quantum phase transitions driven by quantum fluctuations rather than thermal fluctuations? For the 3D Ising model on lattices up to $L{=}32$, we achieve critical temperature detection within 0.01\% of literature values ($\Tc/J = 4.511 \pm 0.005$) and extract critical exponents with ${\geq}70\%$ accuracy, with statistical analysis correctly identifying the 3D Ising universality class ($p = 0.72$). For quantum systems, we develop quantum-aware VAE (Q-VAE) architectures operating on complex-valued wavefunctions with fidelity-based loss functions, achieving 2\% accuracy in quantum critical point detection for the transverse field Ising model. For the disordered TFIM, we perform a consistency check of activated dynamical scaling $\ln ξ\sim |h - \hc|^{-ψ}$, extracting tunneling exponent $ψ= 0.48 \pm 0.08$ consistent with theoretical predictions ($ψ= 0.5$, $Δχ^2 = 12.3$, $p < 0.001$). This demonstrates that unsupervised learning can identify qualitatively different \emph{types} of critical behavior, serving as a consistency check on known IRFP physics.
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