低能量子子空间学习的标度理论
Scaling Theory for Learning Low-Energy Quantum Subspaces
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中文总结 AI 辅助
本文提出低能量子子空间学习的标度理论，证明采样误差与距离和系统尺寸的幂律关系，并用于相变探测。
中文摘要 AI 辅助
相图和能量面需要跨越一族哈密顿量的低能态，但独立求解每个参数点在计算上极其昂贵。一个重要的问题不是单个标记本征态在参数变化下如何表现，而是关于低能扇区的信息有多少包含在一小组参考波函数中。我们表明，物理上相关的对象是孤立的低能子空间本身：它通过简并和能级交叉仍保持良好定义，并且可以从附近采样的态中高效学习。在该子空间中工作，我们证明一种抵消前$q$个非恒定响应阶数的采样模式，对于每个保留能级，产生由$d^{2(q+1)}$限制的能量密度误差，其中$d$是参数空间采样距离。对于$n$个格点上的有能隙局域基态，局域性将此改进为$d^2(nd^2)^q$，从而确定$nd^2$为控制来自采样波函数的高阶信息有效性的自然标度变量。在局域解析区域中，这给出固定参数维度$D$下的采样成本$K=\mathcal{O}([\log(1/\varepsilon)]^D)$。海森堡链和横场伊辛链的数值结果证实了距离标度律和$nd^2$塌缩。除了表征可学习性，标度行为还提供了临界点附近相变的有限尺寸探针。
英文摘要
Phase diagrams and energy surfaces require low-energy states across a family of Hamiltonians, but solving each parameter point independently is prohibitively expensive. An important question is not how a single labeled eigenstate behaves under parameter variation, but how much information about the low-energy sector is contained in a small set of reference wavefunctions. We show that the physically relevant object is the isolated low-energy subspace itself: it remains well defined through degeneracies and level crossings, and it can be learned efficiently from nearby sampled states. Working in this subspace, we prove that a sampling pattern cancelling the first $q$ nonconstant response orders yields an energy-density error bounded by $d^{2(q+1)}$ for every retained level, where $d$ is the parameter-space sampling distance. For a gapped local ground state on $n$ sites, locality sharpens this to $d^2(nd^2)^q$, identifying $nd^2$ as the natural scaling variable governing the effectiveness of higher-order information from sampled wavefunctions. In a local analytic regime this gives a sampling cost $K=\mathcal{O}([\log(1/\varepsilon)]^D)$ at fixed parameter dimension $D$. Numerical results on Heisenberg and transverse-field Ising chains confirm the distance scaling law and the $nd^2$ collapse. Beyond characterizing learnability, the scaling behavior further provides a finite-size probe of phase transitions near criticality.
发表机构
- Department of Physics, Tsinghua University（清华大学物理系）
- Institute of Advanced Study, Tsinghua University（清华大学高等研究院）
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