  Nonlinear variants of quantum mechanics can solve tasks that are impossible
in standard quantum theory, such as perfectly distinguishing nonorthogonal
states. Here we derive the optimal protocol for distinguishing two states of a
qubit using the Gross-Pitaevskii equation, a model of nonlinear quantum
mechanics that arises as an effective description of Bose-Einstein condensates.
Using this protocol, we present an algorithm for unstructured search in the
Gross-Pitaevskii model, obtaining an exponential improvement over a previous
algorithm of Meyer and Wong. This result establishes a limitation on the
effectiveness of the Gross-Pitaevskii approximation. More generally, we
demonstrate similar behavior under a family of related nonlinearities, giving
evidence that the ability to quickly discriminate nonorthogonal states and
thereby solve unstructured search is a generic feature of nonlinear quantum
mechanics.
