  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 $N^{1/4}$. This is a significant, but not unreasonable, improvement
over the $N^{1/2}$ 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.

Quantum Physics