Nonlinear Quantum Search Using the Gross-Pitaevskii Equation

We solve the unstructured search problem in constant time by computing with a physically motivated nonlinearity of the Gross-Pitaevskii type. This speedup comes, however, at the novel expense of increasing the time-measurement precision. Jointly optimizing these resource requirements results in an overall scaling of $N^{1\mathrm{/}4}$. This is a significant, but not unreasonable, improvement over the $N^{1\mathrm{/}2}$ scaling of Grover’s algorithm. Since the Gross-Pitaevskii equation approximates the multi-particle (linear) Schr"odinger equation, for which Grover’s algorithm is optimal, our result leads to a quantum information-theoretic lower bound on the number of particles needed for this approximation to hold, asymptotically.