  Topological data analysis (TDA) is a fast-growing field that utilizes
advanced tools from topology to analyze large-scale data. A central problem in
topological data analysis is estimating the so-called Betti numbers of the
underlying simplicial complex. While the difficulty of this problem has been
established as NP-hard, previous works have showcased appealing quantum
speedup. In this article, we provide an alternative method for estimating Betti
numbers and normalized Betti numbers of given simplicial complex, based on some
recent advances in quantum algorithm, specifically, quantum singular value
transformation. Our method can be faster than the best-known classical method
for finding Betti numbers, and interestingly, it can also find the Betti
numbers of the complement graph to our original one. Comparing to the best
known quantum algorithm, our method generally requires lower depth circuit, in
trade-off for longer running time. Regarding normalized Betti numbers, our
method could match the running time of best-known quantum method in the case of
dense simplices.
