  Quantum algorithms for topological data analysis (TDA) seem to provide an
exponential advantage over the best classical approach while remaining immune
to dequantization procedures and the data-loading problem. In this paper, we
give complexity-theoretic evidence that the central task of TDA -- estimating
Betti numbers -- is intractable even for quantum computers. Specifically, we
prove that the problem of computing Betti numbers exactly is #P-hard, while the
problem of approximating Betti numbers up to multiplicative error is NP-hard.
Moreover, both problems retain their hardness if restricted to the regime where
quantum algorithms for TDA perform best. Because quantum computers are not
expected to solve #P-hard or NP-hard problems in subexponential time, our
results imply that quantum algorithms for TDA offer only a polynomial advantage
in the worst case. We support our claim by showing that the seminal quantum
algorithm for TDA developed by Lloyd, Garnerone and Zanardi achieves a
quadratic speedup over the best known classical approach in asymptotically
almost all cases. Finally, we argue that an exponential quantum advantage can
be recovered if the input data is given as a specification of simplices rather
than as a list of vertices and edges.
