Nonlinear Quantum Search Using the Gross-Pitaevskii Equation

TL;DR

This paper demonstrates a nonlinear quantum search method using the Gross-Pitaevskii equation that achieves faster search times at the cost of higher measurement precision, improving scaling from N^{1/2} to N^{1/4}.

Contribution

It introduces a nonlinear quantum search algorithm based on the Gross-Pitaevskii equation, providing a new speedup over traditional quantum search methods.

Findings

Achieves unstructured search in constant time with nonlinear dynamics.

Optimizing resource trade-offs yields an overall N^{1/4} scaling.

Establishes a lower bound on particle number for the approximation to hold.

Abstract

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.