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Nonlinear Quantum Search

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 $|ψ|^2ψ$. 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 $f(|ψ|^2)ψ$ 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