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A Quantum Algorithm Detecting Concentrated Maps.

Isabel Beichl1, Stephen S Bullock2, Daegene Song3

  • 1National Institute of Standards and Technology, Gaithersburg, MD 20899.

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|April 26, 2016
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Summary
This summary is machine-generated.

This study introduces a quantum algorithm to efficiently distinguish between concentrated and one-to-one functions using quantum oracle calls. The quantum approach offers significant advantages over classical methods for specific function types.

Keywords:
Deutsch-Jozsa algorithmDeutsch’s algorithmconcentrated mapsone-to-one mappingsquantum computationquantum oracleroots of unity

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Area of Science:

  • Quantum computing
  • Theoretical computer science
  • Information theory

Background:

  • Classical algorithms can determine if a function f is one-to-one using N calls to an oracle.
  • A function can be classified as θ-concentrated based on a specific mathematical condition involving a radian angle θ.
  • Distinguishing between these function types is crucial for various computational tasks.

Purpose of the Study:

  • To develop a quantum algorithm for efficiently distinguishing θ-concentrated functions from one-to-one functions.
  • To analyze the performance of the quantum algorithm compared to classical methods.
  • To identify the optimal conditions for the quantum algorithm's outperformance.

Main Methods:

  • Consideration of an arbitrary mapping f: {0, …, N - 1} → {0, …, N - 1} for N = 2^n quantum bits.
  • Development of a quantum algorithm utilizing a quantum oracle function Uf.
  • Analysis of the algorithm's performance in terms of oracle calls and probability of success.
  • Comparison with classical random evaluation methods.

Main Results:

  • The quantum algorithm distinguishes θ-concentrated from one-to-one functions in O(1) calls to the quantum oracle Uf with high probability.
  • For 0 < θ < 0.3301 radians, the quantum algorithm outperforms classical random evaluation.
  • Maximal outperformance is observed at θ = π/8 radians.

Conclusions:

  • A novel quantum algorithm provides a highly efficient method for function property testing.
  • Quantum computation offers a significant advantage over classical approaches for specific problems in function analysis.
  • The algorithm's efficiency is dependent on the degree of function concentration (θ).