  We consider the problem of finding the minimum element in a list of length
$N$ using a noisy comparator. The noise is modelled as follows: given two
elements to compare, if the values of the elements differ by at least $\alpha$
by some metric defined on the elements, then the comparison will be made
correctly; if the values of the elements are closer than $\alpha$, the outcome
of the comparison is not subject to any guarantees. We demonstrate a quantum
algorithm for noisy quantum minimum-finding that preserves the quadratic
speedup of the noiseless case: our algorithm runs in time $\tilde O(\sqrt{N
(1+\Delta)})$, where $\Delta$ is an upper-bound on the number of elements
within the interval $\alpha$, and outputs a good approximation of the true
minimum with high probability. Our noisy comparator model is motivated by the
problem of hypothesis selection, where given a set of $N$ known candidate
probability distributions and samples from an unknown target distribution, one
seeks to output some candidate distribution $O(\varepsilon)$-close to the
unknown target. Much work on the classical front has been devoted to speeding
up the run time of classical hypothesis selection from $O(N^2)$ to $O(N)$, in
part by using statistical primitives such as the Scheff\'{e} test. Assuming a
quantum oracle generalization of the classical data access and applying our
noisy quantum minimum-finding algorithm, we take this run time into the
sublinear regime. The final expected run time is $\tilde O(
\sqrt{N(1+\Delta)})$, with the same $O(\log N)$ sample complexity from the
unknown distribution as the classical algorithm. We expect robust quantum
minimum-finding to be a useful building block for algorithms in situations
where the comparator (which may be another quantum or classical algorithm) is
resolution-limited or subject to some uncertainty.
