拓扑数据分析和局部哈密顿量的归一化持久性问题的复杂性
Complexity of Normalized Persistence Problems for Topological Data Analysis and Local Hamiltonians
- Blackett Laboratory, Imperial College London（帝国理工学院伦敦分校黑克特实验室）
- Department of Computing, Imperial College London（帝国理工学院伦敦分校计算系）
- Yukawa Institute for Theoretical Physics & The Hakubi Center, Kyoto University（京都大学山梨研究所及 Hakubi 中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究归一化持久性问题，证明其变体是$\mathsf{DQC}_1$-难且含于$\mathsf{BQP}$，揭示与局部哈密顿量低能子空间谱量估计复杂性的联系，还介绍$\mathsf{SDQC}_1$刻画精确核归一化问题的硬度。
AI中文摘要：
拓扑数据分析（TDA）是利用拓扑从数据中提取模式的机器学习技术，有展现量子优势的潜力。TDA的关键概念是持久同调。本文引入并研究归一化持久性问题，证明其变体是$\mathsf{DQC}_1$-难且含于$\mathsf{BQP}$，还发现它与局部哈密顿量低能子空间谱量估计复杂性的紧密联系，介绍$\mathsf{SDQC}_1$刻画精确核归一化问题的硬度。
英文摘要：
Topological data analysis (TDA) is a machine learning technique that uses topology to extract patterns from data and has shown the potential to exhibit quantum advantage. A key concept in TDA is persistent homology, which measures the robustness of topological information at different lengthscales. In this paper, we introduce and study the problem of normalized persistence, a practically motivated and easily interpretable version of persistent homology that counts the fraction of holes that persist at different lengthscales. We prove that a variant of normalized persistence is $\mathsf{DQC}_1$-hard and contained in $\mathsf{BQP}$, giving evidence of an exponential quantum speedup for TDA under the standard assumption that $\mathsf{DQC}_1 \not\subseteq \mathsf{BPP}$. These are the first $\mathsf{DQC}_1$-hardness results for clique complexes, making them directly applicable to TDA instances. We also find a close connection between normalized persistence and the complexity of estimating spectral quantities in the low-energy subspace of local Hamiltonians. We study a family of such problems, including a low-energy normalized subtrace and spectral density. We show that these are $\mathsf{DQC}_1$-hard for $O(1)$-local Hamiltonians, strengthening previous results that required log-local interactions. We also introduce a variant of $\mathsf{DQC}_1$ with perfect completeness ($\mathsf{SDQC}_1$) to characterize the hardness of problems normalized by an exact kernel. This includes normalized persistence for $O(1)$-local Hamiltonians, which we show is $\mathsf{SDQC}_1$-hard.
