  In this note, we develop a bounded-error quantum algorithm that makes $\tilde
O(n^{1/4}\varepsilon^{-1/2})$ queries to a Boolean function $f$, accepts a
monotone function, and rejects a function that is $\varepsilon$-far from being
monotone. This gives a super-quadratic improvement compared to the best known
randomized algorithm for all $\varepsilon = o(1)$. The improvement is cubic
when $\varepsilon = 1/\sqrt{n}$.
