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 languageEnglish (US)
Article number012312
JournalPhysical Review A - Atomic, Molecular, and Optical Physics
Volume89
Issue number1
DOIs
StatePublished - Jan 14 2014
Externally publishedYes

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

@article{12c9dd3f5a11486ebc3670f9b6a6f728,
title = "Quantum search with general nonlinearities",
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 Schr{\"o}dinger 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 Schr{\"o}dinger equation describes light propagation in nonlinear Kerr media with defocusing corrections, and the logarithmic nonlinear Schr{\"o}dinger 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 Schr{\"o}dinger 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.",
author = "Meyer, {David A.} and Thomas Wong",
year = "2014",
month = "1",
day = "14",
doi = "10.1103/PhysRevA.89.012312",
language = "English (US)",
volume = "89",
journal = "Physical Review A - Atomic, Molecular, and Optical Physics",
issn = "1050-2947",
publisher = "American Physical Society",
number = "1",

}

TY - JOUR

T1 - Quantum search with general nonlinearities

AU - Meyer, David A.

AU - Wong, Thomas

PY - 2014/1/14

Y1 - 2014/1/14

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84892495198&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84892495198&partnerID=8YFLogxK

U2 - 10.1103/PhysRevA.89.012312

DO - 10.1103/PhysRevA.89.012312

M3 - Article

AN - SCOPUS:84892495198

VL - 89

JO - Physical Review A - Atomic, Molecular, and Optical Physics

JF - Physical Review A - Atomic, Molecular, and Optical Physics

SN - 1050-2947

IS - 1

M1 - 012312

ER -