发表机构
Caltech; Harvard University（加州理工学院; 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
本文提出矩阵乘积态与树张量网络的恰当不可知学习算法，通过不恰当学习压缩目标并引入比较器-对偶压缩，实现多项式副本复杂度与运行时间。
AI 中文摘要
我们建立了矩阵乘积态与树张量网络的恰当不可知学习。给定任意量子态$\rho$的副本，我们的算法返回一个选定键维度的态$|\psi\rangle$，使得$\langle\psi| \rho |\psi\rangle$与模型类中最优值的差距在$\varepsilon$以内，且不假设$\rho$本身属于或能被该类很好地近似。主要思想是利用不恰当学习来压缩混合态目标，将恰当不可知学习简化为针对有限集合的显式指定纯态的优化。然后，我们引入一种比较器-对偶压缩过程，该过程在均匀保持这些目标与所有有界键比较器重叠的同时降低其键维度，且误差与系统大小无关。对于矩阵乘积态，这给出了在系统大小、局域维度、键维度和$1/\varepsilon$上的多项式副本复杂度，以及在固定局域维度、键维度和精度下系统大小上的多项式运行时间。同一框架在有界度树上实现了树张量网络的恰当不可知学习，当其余参数固定时，副本复杂度和运行时间均为系统大小的多项式。
英文摘要
We establish proper agnostic learning of matrix product states and tree tensor networks. Given copies of an arbitrary quantum state $ρ$, our algorithms return a state $|ψ\rangle$ of chosen bond dimension such that $\langleψ| ρ|ψ\rangle$ is within $\varepsilon$ of the optimum over the model class, without assuming that $ρ$ itself belongs to or is well approximated by that class. The main idea is to use improper learning to compress the mixed-state objective, reducing proper agnostic learning to optimization against a finite collection of explicitly specified pure states. We then introduce a comparator-dual compression procedure that reduces the bond dimension of these targets while uniformly preserving their overlaps with all bounded-bond comparators, with an error independent of system size. For matrix product states, this gives polynomial copy complexity in the system size, local dimension, bond dimension, and $1/\varepsilon$, together with polynomial runtime in the system size for fixed local dimension, bond dimension, and accuracy. The same framework yields proper agnostic learning of tree tensor networks on bounded-degree trees, with both copy complexity and runtime polynomial in the system size when the remaining parameters are fixed.
Comments15+37 pages
