# Quantum search with general nonlinearities

David A. Meyer, Thomas Wong

Research output: Contribution to journalArticle

10 Citations (Scopus)

### Abstract

Evolution by the Gross-Pitaevskii equation, which describes Bose-Einstein condensates under certain conditions, solves the unstructured search problem more efficiently than does the Schrödinger equation, because it includes a cubic nonlinearity, proportional to |ψ|2ψ. This is not the only nonlinearity of the form f(|ψ|2)ψ that arises in effective equations for the evolution of real quantum physical systems, however: The cubic-quintic nonlinear Schrödinger equation describes light propagation in nonlinear Kerr media with defocusing corrections, and the logarithmic nonlinear Schrödinger equation describes Bose liquids under certain conditions. Analysis of computation with such systems yields some surprising results; for example, when time-measurement precision is included in the resource accounting, searching a "database" when there is a single correct answer may be easier than searching when there are multiple correct answers. In each of these cases the nonlinear equation is an effective approximation to a multiparticle Schrödinger equation, for search by which Grover's algorithm is optimal. Thus our results lead to quantum information-theoretic bounds on the physical resources required for these effective nonlinear theories to hold, asymptotically.

Original language English (US) 012312 Physical Review A - Atomic, Molecular, and Optical Physics 89 1 https://doi.org/10.1103/PhysRevA.89.012312 Published - Jan 14 2014 Yes

### Fingerprint

nonlinearity
nonlinear equations
resources
defocusing
Bose-Einstein condensates
time measurement
propagation
liquids
approximation

### All Science Journal Classification (ASJC) codes

• Atomic and Molecular Physics, and Optics

### Cite this

Quantum search with general nonlinearities. / Meyer, David A.; Wong, Thomas.

In: Physical Review A - Atomic, Molecular, and Optical Physics, Vol. 89, No. 1, 012312, 14.01.2014.

Research output: Contribution to journalArticle

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