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A concept of controlling Grover diffusion operator: a new approach to solve arbitrary Boolean-based problems.

Ali Al-Bayaty1, Marek Perkowski2

  • 1Department of Electrical and Computer Engineering, Portland State University, Portland, USA. albayaty@pdx.edu.

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Summary

A new controlled-diffusion operator enhances Grover's algorithm for Boolean oracles. This method reliably finds solutions for all logical structures, overcoming limitations of the standard Grover diffusion operator.

Keywords:
Boolean oracleControlled-diffusion operatorGrover diffusion operatorGrover’s algorithmLogical structuresPhase oracle

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Area of Science:

  • Quantum Computing
  • Algorithm Design

Background:

  • Grover's algorithm is a quantum search algorithm.
  • The standard Grover diffusion operator faces limitations with certain Boolean oracle structures.
  • Phase oracles can also exhibit incorrect behavior with the standard operator.

Purpose of the Study:

  • To design a novel controlled-diffusion operator for Grover's algorithm.
  • To address the shortcomings of the standard Grover diffusion operator for arbitrary Boolean and Phase oracles.
  • To ensure reliable solution searching across diverse oracle logical structures.

Main Methods:

  • Development of a controlled-diffusion operator utilizing output qubit states.
  • Reflection mechanism based on Boolean decisions, independent of phase kickback.
  • Mathematical modeling and experimental validation.

Main Results:

  • Demonstrated failure of the Grover diffusion operator on specific Boolean and Phase oracles.
  • Examples of problematic oracles include POS, SOP, ESOP, CSP-SAT, and XOR-SAT structures.
  • The proposed controlled-diffusion operator successfully identified all solutions for tested oracles.

Conclusions:

  • The controlled-diffusion operator offers a robust improvement over the standard Grover diffusion operator.
  • This new operator guarantees accurate solutions for all Boolean oracles, irrespective of their logical complexity.
  • The findings have significant implications for quantum search algorithms and solving complex logical problems.