  Evolution by the Gross-Pitaevskii equation, which describes Bose-Einstein
condensates under certain conditions, solves the unstructured search problem
more efficiently than does the Schr\"odinger equation, because it includes a
cubic nonlinearity, proportional to $|\psi|^2\psi$. This is not the only
nonlinearity of the form $f(|\psi|^2)\psi$ that arises in effective equations
for the evolution of real quantum physical systems, however: The cubic-quintic
nonlinear Schr\"odinger equation describes light propagation in nonlinear Kerr
media with defocusing corrections, and the logarithmic nonlinear Schr\"odinger
equation describes Bose liquids under certain conditions. Analysis of
computation with such systems yields some surprising results; for example, when
time-measurement precision is included in the resource accounting, searching a
"database" when there is a single correct answer may be easier than searching
when there are multiple correct answers. In each of these cases the nonlinear
equation is an effective approximation to a multi-particle Schr\"odinger
equation, for search by which Grover's algorithm is optimal. Thus our results
lead to quantum information-theoretic bounds on the physical resources required
for these effective nonlinear theories to hold, asymptotically.
