发表机构
Institute of Advanced Computer Science, Leiden University; School of Physics and Astronomy, Monash University; Faculty of Science and Computer, Universitas Kristen Immanuel; Department of Computer Science, University of Oxford; Department of Computer Science, Royal Holloway University of London; Institut für Theoretische Physik, Leibniz Universität Hannover; Matematikos ir informatikos fakultetas, Vilniaus universiteto（莱顿大学高级计算机科学研究所; 蒙纳士大学物理与天文学院; 伊曼纽尔基督教大学科学与计算机学院; 牛津大学计算机科学系; 伦敦皇家霍洛威学院计算机科学系; 汉诺威莱布尼茨大学理论物理研究所; 维尔纽斯大学数学与信息学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出首个可证明正确的量子算法，利用对数伪行列式估计整数同调挠阶，并给出含挠基准与查询复杂度优势。
AI 中文摘要
Betti数是拓扑数据分析的核心，但无法捕捉整数同调中的挠。挠可以区分具有相同Betti数的拓扑结构，经典拓扑数据分析中的例子表明，挠可以影响从数据推断出的拓扑。相关的挠敏感量在基于图的机器学习中也有实际应用，包括生物医学应用。我们研究量子谱方法如何获取这些额外信息。通过对非零边界-拉普拉斯特征值取对数求和得到的对数伪行列式，总结了单纯复形的谱信息。在明确的拓扑假设下，这些量的交替组合给出了高阶临界群的大小，即图沙堆模型群的高维类比。在更强的认证条件下，一个相关构造恢复了同调本身中挠子群的大小。据我们所知，这是第一个可证明正确的整数同调挠阶的量子估计器。我们给出了一种新的量子算法来估计这些量，其显式依赖于输入的查询方式、所需精度和谱间隙。平衡多部团复形的精确谱提供了有利的间隙，一个相关构造给出了一个显式的含挠基准。在紧凑查询访问下，该算法无需显式构造指数大的边界矩阵。对于一个受限的含挠族，一个定制的振幅估计例程在估计对数伪行列式时，与具有可比访问权限的任何随机经典算法相比，使用的查询次数呈二次减少。估计完整的、未归一化的量到固定加性误差仍然与候选空间维度$D$成比例。
英文摘要
Betti numbers are central to topological data analysis but fail to capture torsion in integral homology. Torsion can distinguish topological structures with identical Betti numbers, and examples from classical topological data analysis show that it can affect the topology inferred from data. Related torsion-sensitive quantities are also known to have practical applications in graph-based machine learning, including biomedical applications. We investigate how quantum spectral methods can access this additional information. Log pseudodeterminants, obtained by summing the logarithms of the nonzero boundary-Laplacian eigenvalues, summarize spectral information about a simplicial complex. Under explicit topological assumptions, an alternating combination of these quantities gives the size of a higher critical group, the higher-dimensional analogue of a graph sandpile group. With stronger certification, a related construction recovers the size of the torsion subgroup in homology itself. This is, to our knowledge, the first provably correct quantum estimator of integral-homology torsion order. We give a new quantum algorithm for estimating these quantities, with explicit dependence on how the input is queried, the required accuracy, and the spectral gaps. Exact spectra for balanced multipartite clique complexes provide favorable gaps, and a related construction gives an explicit torsion-bearing benchmark. With compact query access, the algorithm need not construct the exponentially large boundary matrices explicitly. For a restricted torsion-bearing family, a tailored amplitude-estimation routine uses quadratically fewer queries than any randomized classical algorithm given comparable access when estimating the log pseudodeterminant. Estimating the full, unnormalized quantity to fixed additive error still costs in proportion to the candidate-space dimension $D$.
