We introduce several new quantum algorithms for estimating homological invariants, specifically Betti numbers and persistent Betti numbers, of a simplicial complex given via a structured classical input. At the core of our algorithm lies the ability to efficiently construct the block-encoding of Laplacians (and persistent Laplacians) based on the classical description of the given complex. From such block-encodings, Betti numbers (and persistent Betti numbers) can be estimated. The complexity of our method is polylogarithmic in the number of simplices in both simplex-sparse and simplex-dense regimes, thus offering an advantage over existing works. Moreover, prior quantum algorithms based on spectral methods incur significant overhead due to their reliance on estimating the kernel of combinatorial Laplacians, particularly when the Betti number is small. We introduce a new approach for estimating Betti numbers based on homology tracking and homology property testing, which enables exponential quantum speedups over both classical and prior quantum approaches under sparsity and structure assumptions. We further initiate the study of homology triviality and equivalence testing as natural property testing problems in topological data analysis, and provide efficient quantum algorithms with time complexity nearly linear in the number of simplices when the rank of the boundary operator is large. In addition, we develop a cohomological approach based on block-encoded projections onto cocycle spaces, enabling rank-independent testing of homology equivalence. This yields the first quantum algorithms for constructing and manipulating r-cocycles in time polylogarithmic in the size of the complex. Together, these results establish a new direction in quantum topological data analysis and demonstrate that computing topological invariants can serve as a fertile ground for provable quantum advantage.

翻译：我们提出了几种新的量子算法，用于估计单纯复形（给定于结构化经典输入）的同调不变量，特别是贝蒂数和持续贝蒂数。算法的核心在于，基于给定复形的经典描述，能够高效构建拉普拉斯算子（及持续拉普拉斯算子）的块编码。通过这些块编码，可以估计贝蒂数（及持续贝蒂数）。在单形稀疏和单形密集两种情形下，我们的方法复杂度均与单形数量呈多对数关系，相比现有工作具有优势。此外，先前基于谱方法的量子算法因依赖组合拉普拉斯算子核的估计而引入显著开销，尤其在贝蒂数较小时。我们提出一种基于同调追踪和同调性质测试的贝蒂数估计新方法，在稀疏性和结构假设下，相比经典和先前量子方法可实现指数级量子加速。我们进一步将同调平凡性测试和等价测试作为拓扑数据分析中的自然性质测试问题，并提供了高效量子算法：当边界算子秩较大时，时间复杂度近似线性于单形数量。此外，我们开发了一种基于上循环空间块编码投影的上同调方法，实现了与秩无关的同调等价性测试。这首次实现了在复形尺寸的多对数时间内构建与操作r-上循环的量子算法。这些成果共同开辟了量子拓扑数据分析的新方向，表明计算拓扑不变量可为可证量子优势提供丰饶土壤。