  Although quant mech is linear, there are nevertheless quant sys with multiple
interacting particles in which the effective evo of a single particle is
governed by a nonlinear eq. This includes Bose-Einstein condensates, which are
gov by the Gross-Pitaevskii eq (GPE), which is a cubic nonlin Schrodinger eq
(NLSE) with a term propto $|\psi|^2\psi$. Evo by this eq solves the unstruct
search prob in const time, but at the novel expense of increasing the
time-measurement precision. Jointly optimizing these resources results in an
overall scaling of $N^{1/4}$, which is a significant, but not unreasonable,
improvement over the $N^{1/2}$ scaling of Grover's algo. Since the GPE
effectively approx the multi-particle Schrodinger eq, for which Grover's algo
is optimal, our result leads to a quant info-theoretic bound on the num of
particles needed for this approx to hold, asymp. The GPE is not the only nonlin
of the form $f(|\psi|^2)\psi$ that arises in effective eqs for the evo of real
quant phys sys, however: The cubic-quintic NLSE describes light propagation in
nonlin Kerr media with defocusing corrections, and the log NLSE describes Bose
liquids under certain cond. Analysis of comput with such sys yields some
surprising results; e.g., when time-measurement precision is included in the
resource accounting, searching a "database" when there is a single correct ans
may be easier than searching when there are multiple correct ans. These results
lead to quant info-theoretic bounds on the phys resources required for these
effective nonlin theories to hold, asymp. Further, strongly reg graphs, which
have no global sym, are sufficiently complete for quant search on them to asymp
behave like unstruct search. Certain suff complete graphs retain the improved
runtime and resource scalings for some nonlin, so our scheme for nonlin, analog
quant comput retains its benefits even when some struct is introduced.
