  Topological invariants of a dataset, such as the number of holes that survive
from one length scale to another (persistent Betti numbers) can be used to
analyse and classify data in machine learning applications. We present an
improved quantum algorithm for computing persistent Betti numbers, and provide
an end-to-end complexity analysis. Our approach provides large polynomial time
improvements, and an exponential space saving, over existing quantum
algorithms. Subject to gap dependencies, our algorithm obtains an almost
quintic speedup in the number of datapoints over rigorous state-of-the-art
classical algorithms for calculating the persistent Betti numbers to constant
additive error - the salient task for applications. However, this may be
reduced to closer to quadratic when compared against heuristic classical
methods and observed scalings. We discuss whether quantum algorithms can
achieve an exponential speedup for tasks of practical interest, as claimed
previously. We conclude that there is currently no evidence that this is the
case.

Quantum Physics