**Using fewer Qubits in Shor's Factorization Algorithm via Simultaneous Diophantine Approximation**

*Jean-Pierre Seifert*

**Abstract: **While quantum computers might speed up in principle
certain computations dramatically, in pratice, though
quantum computing technology is still in its infancy.
Even we cannot clearly envison 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 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 $N^2$, our simultaneous Diophantine approximation method computes in polynomial time unusually good approximations to known fractions with a denominator of size $N^{1+\varepsilon}$, where $\varepsilon$ 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.

**Category / Keywords: **foundations / factoring, quantum computing

**Date: **received 18 Jul 2000

**Contact author: **Jean-Pierre Seifert at Infineon com

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**Version: **20000718:170508 (All versions of this report)

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