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Basics of Multivariate Analysis in Neuroimaging Data
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Quantum algorithm for multivariate polynomial interpolation.

Jianxin Chen1, Andrew M Childs2,3,1, Shih-Han Hung2,1

  • 1Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD 20742, USA.

Proceedings. Mathematical, Physical, and Engineering Sciences
|February 14, 2018
PubMed
Summary
This summary is machine-generated.

Quantum algorithms significantly reduce the queries needed for multivariate polynomial interpolation. This research demonstrates a substantial speed-up compared to classical methods, particularly for larger polynomial degrees and variables.

Keywords:
polynomial interpolationquantum algorithmsquery complexity

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Area of Science:

  • Quantum Computing
  • Computational Complexity Theory
  • Algebraic Geometry

Background:

  • Determining polynomial coefficients is a fundamental problem in computation.
  • Classical algorithms for multivariate polynomial interpolation face high query complexity.
  • Understanding the quantum query complexity offers insights into potential computational advantages.

Purpose of the Study:

  • To investigate the quantum query complexity for multivariate polynomial interpolation.
  • To develop and analyze quantum algorithms for this problem over finite fields.
  • To compare the efficiency of quantum versus classical approaches.

Main Methods:

  • Development of quantum algorithms tailored for polynomial interpolation.
  • Analysis of query complexity over specific finite fields (F_2, F_3, and large q).
  • Comparison of quantum query counts against established classical bounds.

Main Results:

  • Achieved probability 1 with specific query counts for F_2 and F_3.
  • Established query complexity for large field order q as ⌈(d/(n+d))(n+d choose d)⌉.
  • Demonstrated significant speed-ups over classical methods, with factors up to n+1.

Conclusions:

  • Quantum algorithms offer a considerable advantage for multivariate polynomial interpolation.
  • The gap between quantum and classical query complexity is larger than in the univariate case.
  • Further research is needed to confirm conjectured optimal query counts for certain fields.