发表机构
C. N. Yang Institute for Theoretical Physics, State University of New York at Stony Brook; Institute of Applied Mathematics, University of Economics Ho Chi Minh City; School of Mathematical and Physical Sciences, Macquarie University; Department of Natural Sciences, Scripps and Pitzer Colleges, Claremont Colleges Consortium（纽约州立大学石溪分校陈省身理论物理研究所; 胡志明市经济大学应用数学研究所; 麦考瑞大学数理科学学院; 克莱蒙特学院联盟斯克里普斯学院和皮泽学院自然科学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明判定图团复形同调群中素数阶挠性的存在是NP困难的，并提出一种量子算法作为单侧挠性见证，在特定条件下实现近二次量子加速，揭示了超越贝蒂数的积分同调的计算复杂性。
AI 中文摘要
近期进展揭示了量子计算与拓扑数据分析（TDA）之间的相互作用。大多数量子TDA聚焦于贝蒂数，它刻画数据集的连通性和“空洞”。然而，同调还包含以挠性形式存在的额外信息：一个非平凡循环在有限次重复后可能变为平凡，揭示了循环如何组合和相互缠绕的全局约束。除生物分子研究中的应用外，挠性还出现在物理场景中，包括同调量子转子码、离散荷和规范扇区。我们从经典和量子两个角度研究挠性。给定图$G$及其团复形$K = \mathrm{Cl}(G)$，我们首先证明，对于固定的$r$和素数$p$，判定$H_r(K,\mathbb{Z})$是否包含$p$-挠性是NP困难的。作为推论，当同调转子码由$G$指定时，判定该码是否具有给定阶数的有限维逻辑扇区是NP困难的。我们讨论了相关问题，包括Bockstein同态、Smith标准型、格饱和和上同调。其次，对于有限素数集$P$，我们开发了一种量子算法，作为单侧挠性见证。对于固定的$r$，它输出WITNESS或INCONCLUSIVE，即WITNESS证明$H_r(K,\mathbb{Z})$或$H_{r-1}(K,\mathbb{Z})$对某个$p\in P$包含$p$-挠性，而INCONCLUSIVE则对其存在与否不作断言。我们确定了一个算法在相同输入模型下相对于相应经典算法实现近二次量子加速的机制。我们的NP困难结果补充了最近关于估计贝蒂数的困难性结果，为量子TDA增添了复杂度理论视角。这些结果共同表明，积分同调，超越其贝蒂数，在计算上可能具有挑战性。
英文摘要
Recent advances have revealed an interplay between quantum computing and topological data analysis (TDA). Most quantum TDA has focused on Betti numbers, which characterize the connectivity and ``holes'' of a dataset. Homology, however, contains additional information in the form of torsion: a nontrivial cycle can become trivial after being repeated finitely, revealing global constraints on how cycles combine and wrap around one another. Beyond applications in biomolecular studies, torsion appears in physical settings including homological quantum rotor codes, discrete charges, and gauge sectors. We study torsion from both classical and quantum perspectives. Given a graph $G$ and its clique complex $K = \mathrm{Cl}(G)$, we first prove that, for fixed $r$ and prime $p$, deciding whether $H_r(K,\mathbb{Z})$ contains $p$-torsion is NP-hard. As a corollary, when a homological rotor code is specified by $G$, deciding whether the code has a finite-dimensional logical sector of a given order is NP-hard. We discuss related problems, including the Bockstein homomorphism, Smith normal form, lattice saturation, and cohomology. Second, for a finite set of primes $P$, we develop a quantum algorithm that serves as a one-sided torsion witness. For fixed $r$, it outputs WITNESS or INCONCLUSIVE, i.e. WITNESS certifies that either $H_r(K,\mathbb{Z})$ or $H_{r-1}(K,\mathbb{Z})$ contains $p$-torsion for some $p\in P$ while INCONCLUSIVE makes no claim about its presence or absence. We identify a regime in which the algorithm achieves a near-quadratic quantum speedup over the corresponding classical algorithm under the same input model. Our NP-hardness result complements recent hardness results for estimating Betti numbers, adding a complexity-theoretic perspective to quantum TDA. Together, these results demonstrate that integral homology, beyond its Betti numbers, can be computationally challenging.
