In this note, we give tighter bounds on the complexity of the bipartite unique perfect matching problem, bipartite-UPM. We show that the problem is in C=L and in NL , both subclasses of NC. We also consider the (unary) weighted version of the problem. We show that testing uniqueness of the minimum-weight perfect matching problem for bipartite graphs is in L= and in NL. Furthermore, we show that bipartite-UPM is hard for NL.