Optimal state discrimination and unstructured search in nonlinear quantum mechanics

TL;DR

This paper derives the optimal protocol for state discrimination in nonlinear quantum mechanics using the Gross-Pitaevskii equation and demonstrates an exponential speedup for unstructured search, highlighting the power and limitations of nonlinear quantum models.

Contribution

It introduces the optimal state discrimination protocol in nonlinear quantum mechanics and shows an exponential speedup for unstructured search within this framework.

Findings

Optimal protocol for qubit state discrimination derived.

Exponential improvement in unstructured search algorithm.

Demonstrates generic features of nonlinear quantum mechanics.

Abstract

Nonlinear variants of quantum mechanics can solve tasks that are impossible in standard quantum theory, such as perfectly distinguishing nonorthogonal states. Here we derive the optimal protocol for distinguishing two states of a qubit using the Gross-Pitaevskii equation, a model of nonlinear quantum mechanics that arises as an effective description of Bose-Einstein condensates. Using this protocol, we present an algorithm for unstructured search in the Gross-Pitaevskii model, obtaining an exponential improvement over a previous algorithm of Meyer and Wong. This result establishes a limitation on the effectiveness of the Gross-Pitaevskii approximation. More generally, we demonstrate similar behavior under a family of related nonlinearities, giving evidence that the ability to quickly discriminate nonorthogonal states and thereby solve unstructured search is a generic feature of nonlinear quantum mechanics.