AI 中文总结
研究临界状态下参数化量子演化中，基于张量网络有限资源标度为近似分配涌现关联长度$\xi_D$，用其增长定义指数$\kappa$衡量架构效率，应用于横向场伊辛模型，比较不同近似及层组织，表明其可作红外分辨率尺度及架构设计指导。
AI 中文摘要
变分近似是量子多体物理的基石，利用有限资源为复杂基态提供紧凑近似。近期量子技术进展引入了基于分层参数化演化的新类别。评估其能否表示临界基态具有挑战性，因为关联跨越所有长度尺度，且有限电路深度限制了其扩展范围。基于张量网络的有限资源标度，我们为每个近似分配一个涌现关联长度$\xi_D$，它随细化参数$D$的增长$\xi_D \propto D^\kappa$定义了一个指数$\kappa$，用于衡量架构将资源转化为长距离关联的效率。将此框架应用于临界横向场伊辛模型，以$D$为参数化演化的电路深度，我们比较了具有最近邻和长程生成器以及生成器单独或组合作用的层的近似。所有近似都与代数增长兼容，但拟合指数范围从$\kappa\simeq1$到$\kappa\simeq3$。指数相互作用给出最大指数，而幂律相互作用接近最近邻行为，表明仅长程支持没有标度优势。组合生成器层通常优于可分的层，所以层组织与相互作用范围同样重要。最后，准粒子分析表明有限深度在负责长距离关联的低能模式周围留下宽度为$\xi_D^{-1}$的未解决窗口。涌现关联长度因此充当红外分辨率尺度，为临界态制备提供基准并为设计资源高效的变分架构提供指导。
英文摘要
Variational ansätze are a cornerstone of quantum many-body physics, providing compact approximations to complex ground states using finite resources. Recent quantum-technology advances have introduced a new class based on layered parameterized evolutions. Assessing whether they can represent critical ground states is challenging: correlations span all length scales, while finite circuit depth limits how far they extend. Building on finite-resource scaling from tensor networks, we assign each ansatz an emergent correlation length $ξ_D$, the longest range over which it faithfully captures critical correlations. Its growth with refinement parameter $D$, $ξ_D \propto D^κ$, defines an exponent $κ$ measuring how efficiently an architecture converts resources into long-distance correlations. Applying this framework to the critical transverse-field Ising model, with $D$ the circuit depth of parameterized evolutions, we compare ansätze with nearest-neighbor and long-range generators, and layers where generators act separately or combined. All ansätze are compatible with algebraic growth, but fitted exponents range from $κ\simeq1$ to $κ\simeq3$. Exponential interactions give the largest exponents, while power-law interactions stay close to nearest-neighbor behavior, showing long-range support alone gives no scaling advantage. Combined-generator layers generally outperform separable ones, so layer organization matters alongside interaction range. Finally, a quasiparticle analysis shows finite depth leaves an unresolved window of width $ξ_D^{-1}$ around the low-energy modes responsible for long-distance correlations. The emergent correlation length thus acts as an infrared resolution scale, providing a benchmark for critical-state preparation and a guide for designing resource-efficient variational architectures.
Comments24 pages, 9 figures
