We present new quantum algorithms for estimating homological invariants, specifically Betti and persistent Betti numbers, of a simplicial complex given through structured classical data. Our approach efficiently constructs block-encodings of (persistent) Laplacians, enabling estimation via stochastic rank methods with complexity polylogarithmic in the number of simplices across both sparse and dense regimes. Unlike prior spectral algorithms that suffer when Betti numbers are small, we introduce homology tracking and property testing techniques achieving exponential speedups under natural sparsity and structure assumptions. We also formulate homology triviality and equivalence testing as property testing problems, giving nearly linear-time quantum algorithms when the boundary rank is large. A cohomological formulation further yields rank-independent testing and polylog-time manipulation of $r$-cocycles via block-encoded projections. These results open a new direction in quantum topological data analysis and demonstrate provable quantum advantages in computing topological invariants.

翻译：我们提出了新的量子算法，用于估计由结构化经典数据给出的单纯复形的同调不变量，特别是贝蒂数和持久贝蒂数。我们的方法高效构建了（持久）拉普拉斯算子的块编码，使得通过随机秩方法进行估计成为可能，其复杂度在稀疏和稠密情形下均与单纯形数量的多对数相关。与先前在贝蒂数较小时性能受限的谱算法不同，我们引入了同调追踪和性质检验技术，在自然的稀疏性和结构假设下实现了指数级加速。我们还将同调平凡性与等价性检验形式化为性质检验问题，当边界秩较大时给出了近线性时间的量子算法。通过上同调表述进一步实现了与秩无关的检验，并借助块编码投影实现了$r$-上闭链的多对数时间操作。这些结果为量子拓扑数据分析开辟了新方向，并展示了在计算拓扑不变量方面可证明的量子优势。