We introduce a new quantum algorithm for computing the Betti numbers of a simplicial complex. In contrast to previous quantum algorithms that work by estimating the eigenvalues of the combinatorial Laplacian, our algorithm is an instance of the generic Incremental Algorithm for computing Betti numbers that incrementally adds simplices to the simplicial complex and tests whether or not they create a cycle. In contrast to existing quantum algorithms for computing Betti numbers that work best when the complex has close to the maximal number of simplices, our algorithm works best for sparse complexes. To test whether a simplex creates a cycle, we introduce a quantum span-program algorithm. We show that the query complexity of our span program is parameterized by quantities called the effective resistance and effective capacitance of the boundary of the simplex. Unfortunately, we also prove upper and lower bounds on the effective resistance and capacitance, showing both quantities can be exponentially large with respect to the size of the complex, implying that our algorithm would have to run for exponential time to exactly compute Betti numbers. However, as a corollary to these bounds, we show that the spectral gap of the combinatorial Laplacian can be exponentially small. As the runtime of all previous quantum algorithms for computing Betti numbers are parameterized by the inverse of the spectral gap, our bounds show that all quantum algorithms for computing Betti numbers must run for exponentially long to exactly compute Betti numbers. Finally, we prove some novel formulas for effective resistance and effective capacitance to give intuition for these quantities.

翻译：我们提出了一种用于计算单纯复形贝蒂数的新型量子算法。与以往通过估计组合拉普拉斯算子特征值来工作的量子算法不同，我们的算法是通用增量算法的一个实例，它通过逐步向单纯复形中添加单纯形并测试是否形成环来计算贝蒂数。现有量子算法在复形接近最大单纯形数时表现最佳，而我们的算法则适用于稀疏复形。为了测试单纯形是否形成环，我们提出了一种量子跨度程序算法。我们证明，该跨度程序的查询复杂度由单纯形边界的有效电阻和有效电容这两个参数量化。遗憾的是，我们还推导了有效电阻和电容的上界和下界，证明这两个量可能相对于复形规模呈指数级增长，这意味着我们的算法需要指数级运行时间才能精确计算贝蒂数。然而，作为这些界的推论，我们证明组合拉普拉斯算子的谱隙可能呈指数级缩小。由于以往所有计算贝蒂数的量子算法的运行时间均以谱隙倒数为参量，我们的界表明所有量子算法都需要指数级时间才能精确计算贝蒂数。最后，我们推导了有效电阻和有效电容的一些新公式，以直观展示这些量的性质。