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Factoring 51 and 85 with 8 qubits.

Michael R Geller1, Zhongyuan Zhou

  • 1Department of Physics and Astronomy, University of Georgia, Athens, Georgia 30602, USA.

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|October 29, 2013
PubMed
Summary
This summary is machine-generated.

This study simplifies Shor's algorithm for factoring specific composite numbers using Fermat primes. Simplified quantum circuits require fewer qubits and lower precision for efficient prime factorization.

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Area of Science:

  • Quantum computing
  • Number theory
  • Cryptography

Background:

  • Shor's algorithm is a quantum algorithm for integer factorization.
  • Factoring large numbers is computationally intensive for classical computers.
  • Fermat primes are primes of the form 2^(2^n) + 1.

Purpose of the Study:

  • To construct simplified quantum circuits for Shor's order-finding algorithm.
  • To leverage the properties of composite numbers formed by Fermat primes for efficient factorization.
  • To reduce the resource requirements for quantum factorization.

Main Methods:

  • Developing simplified quantum circuits tailored for specific composite numbers.
  • Utilizing the property that orders modulo N are powers of 2 for composites of Fermat primes.
  • Implementing modular exponentiation with a reduced number of CNOT gates.

Main Results:

  • Demonstrated prime factorization of 51 and 85 using only 8 qubits.
  • Achieved factorization with a modular exponentiation circuit requiring a maximum of four CNOT gates.
  • Showcased significant reduction in phase estimation precision requirements.

Conclusions:

  • Simplified quantum circuits offer a more efficient approach to Shor's algorithm for specific composite numbers.
  • Composites of Fermat primes present a tractable class for quantum factorization.
  • This work paves the way for practical quantum factorization with fewer resources.