We analyze the quantum query complexity of sorting under partial information.
In this problem, we are given a partially ordered set P and are asked to
identify a linear extension of P using pairwise comparisons. For the standard
sorting problem, in which P is empty, it is known that the quantum query
complexity is not asymptotically smaller than the classical
information-theoretic lower bound. We prove that this holds for a wide class of
partially ordered sets, thereby improving on a result from Yao (STOC'04)