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)


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.


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. ABO.
    D. Aharonov, M. Ben-Or, “Fault-tolerant quantum computing with constant error”, Proc. of the 29th Ann. ACM Symp. on Theory of Comp., pp. 176–188, 1997.Google Scholar
  2. Cas.
    J. W. S. Cassels, An Introduction to Diophantine Approximations, Cambridge University Press, Cambridge, 1957.Google Scholar
  3. CZ.
    I. J. Cirac, P. Zoller, “Quantum computations with cold trapped ions”, Phys. Rev. Let. 74:4091–4094, 1995.CrossRefGoogle Scholar
  4. CH+._I. L. Chuang, L. M. K. Vandersypen, X. Zhou, D. W. Leung, S. Lloyd, “Experimental realization of a quantum algorithm”, Nature 393:143–146, 1998.CrossRefGoogle Scholar
  5. Gru.
    J. Gruska, Quantum Computing, McGraw-Hill, London, 1999.Google Scholar
  6. HSS.
    J. Håstad, A. W. Schrift, A. Shamir, “The discrete logarithm modulo a composite hides O(n) bits”, J. Comp. Sys. Sci. 47:376–404, 1993.zbMATHCrossRefGoogle Scholar
  7. K.
    A. Y. Kitaev, “Quantum measurements and the Abelian stabilizer problem”, Technical report, quant-ph/9511026, 1995.Google Scholar
  8. Lag1.
    J. C. Lagarias, “Some new results in simultaneous diophantine approximation”, Queen’s Pap. Pure Appl. Math. 54:453–474, 1980.Google Scholar
  9. Lag2.
    J. C. Lagarias, “Best simultaneous Diophantine approximations. I: Growth rates of best approximation denominators”, Trans. Am. Math. Soc. 272:545–554, 1982.zbMATHCrossRefMathSciNetGoogle Scholar
  10. Lag3.
    J. C. Lagarias, “Best simultaneous Diophantine approximations. II: Behaviour of consecutive best approximations”, Pacific J. Math. 102:61–88, 1982.zbMATHMathSciNetGoogle Scholar
  11. Lag4.
    J. C. Lagarias, “The computational complexity of simultaneous Diophantine approximation problems”, SIAM J. Computing 14:196–209, 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  12. LH.
    J. Lagarias, J. Håstad, “Simultaneous diophantine approximation of rationals by rationals”, J. Number Theory 24:200–228, 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  13. Leg.
    A. M. Legendre, Essai sur la th_eorie des nombres, J. B. M. Duprat, Paris, 1798.Google Scholar
  14. Lov.
    L. Lovasz, An Algorithmic Theory of Graphs, Numbers and Convexity, SIAM Publications, Philadelphia, 1986.Google Scholar
  15. M.
    M. Mosca, “Quantum searching, counting and amplititude modification by eigenvector analysis”, Proc. of the MFCS’98 Workshop on Randomized Algorithms, pp. 90–100, 1998.Google Scholar
  16. ME.
    M. Mosca, A. Ekert, “The hidden subgroup problem and eigenvalue estimation on a quantum computer”, Proc. of the 1st NASA International Conference on Quantum Computing and Quantum Communication, 1998.Google Scholar
  17. PP.
    S. Parker, M. B. Plenio, “Efficient factorization with a single pure qubit”, Technical report, quant-ph/0001066, 2000.Google Scholar
  18. Rit.
    H. Ritter, Zufallsbits basierend auf dem diskreten Logarithmus, Master Thesis, University of Frankfurt, Dept. of Math., 1992.Google Scholar
  19. RSA.
    R. Rivest, A. Shamir, L. Adleman, “A method for obtaining digital signatures and public-key cryptosystems”, Comm. of the ACM 21:120–126, 1978.zbMATHCrossRefMathSciNetGoogle Scholar
  20. RS.
    C. Rössner, J.-P. Seifert, “Approximating good simultaneous diophantine approximations is almost NP-hard”, Proc. 21st Symposium on Mathematical Foundations of Computer Science, pp. 494–504, 1996.Google Scholar
  21. Sch.
    A. Schrijver, An Introduction to Linear and Integer Programming, John Wiley & Sons, New York, 1986.Google Scholar
  22. Kan.
    B. E. Kane, “Silicon based quantum computation”, Technical report, quant-ph/0003031, 2000.Google Scholar
  23. Knu.
    D. E. Knuth, The Art of Computer Programming, Vol.2: Seminumerical Algorithms, 3rd ed., Addison-Wesley, Reading MA, 1999.Google Scholar
  24. Sha.
    A. Shamir, “A polynomial-time algorithm for breaking the basic Merkle-Hellman cryptosystem”, IEEE Trans. Inf. Theory IT-30:699–704, 1984.CrossRefMathSciNetGoogle Scholar
  25. Sho.
    P. Shor, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer”, SIAM J. Computing 26:1484–1509, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  26. Z.
    C. Zalka, “Fast version of Shor’s quantum factoring algorithm”, Technical report, quant-ph/9806084, 1998.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

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

Personalised recommendations