  We looked into the algorithm for calculating Betti numbers presented by
Lloyd, Garnerone, and Zanardi (LGZ). We present a new algorithm in the same
spirit as LGZ with the intent of clarifying quantum algorithms for computing
Betti numbers. Our algorithm is simpler and slightly more efficient than that
presented by LGZ. We present a thorough analysis of our algorithm, pointing out
reasons that both our algorithm and that presented by LGZ do not run in
polynomial time for most inputs. However, the algorithms do run in polynomial
time for calculating an approximation of the Betti number to polynomial
multiplicative error, when applied to some class of graphs for which the Betti
number is exponentially large.
