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Using Fewer Qubits in Shor’s Factorization Algorithm via Simultaneous Diophantine Approximation

  • Jean-Pierre Seifert
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2020)

Abstract

While quantum computers might speed up in principle certain computations dramatically, in practice, though quantum computing technology is still in its infancy. Even we cannot clearly envision at present what the hardware of that machine will be like. Nevertheless, we can be quite confident that it will be much easier to build any practical quantum computer operating on a few number of quantum bits rather than one operating on a huge number of quantum bits. It is therefore of big practical impact to use the resource of quantum bits very spare, i.e., to find quantum algorithms which use as few as possible quantum bits. Here, we present a method to reduce the number of actually needed qubits in Shor’s algorithm to factor a composite number N. Exploiting the inherent probabilism of quantum computation we are able to substitute the continued fraction algorithm to find a certain unknown fraction by a simultaneous Diophantine approximation. While the continued fraction algorithm is able to find a Diophantine approximation to a single known fraction with a denominator greater than N2, our simultaneous Diophantine approximation method computes in polynomial time unusually good approximations to known fractions with a denominator of size N1+∈, where ∈ is allowed to be an arbitrarily small positive constant. As these unusually good approximations are almost unique we are able to recover an unknown denominator using fewer qubits in the quantum part of our algorithm.

Keywords

Quantum Algorithm Rational Vector Diophantine Approximation Small Positive Constant Composite Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Jean-Pierre Seifert
    • 1
  1. 1.Infineon TechnologiesSecurity & ChipCard ICsMunichGermany

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