Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance


The number of steps any classical computer requires in order to find the prime factors of an l-digit integer N increases exponentially with l, at least using algorithms known at present1. Factoring large integers is therefore conjectured to be intractable classically, an observation underlying the security of widely used cryptographic codes1,2. Quantum computers3, however, could factor integers in only polynomial time, using Shor's quantum factoring algorithm4,5,6. Although important for the study of quantum computers7, experimental demonstration of this algorithm has proved elusive8,9,10. Here we report an implementation of the simplest instance of Shor's algorithm: factorization of N = 15 (whose prime factors are 3 and 5). We use seven spin-1/2 nuclei in a molecule as quantum bits11,12, which can be manipulated with room temperature liquid-state nuclear magnetic resonance techniques. This method of using nuclei to store quantum information is in principle scalable to systems containing many quantum bits13, but such scalability is not implied by the present work. The significance of our work lies in the demonstration of experimental and theoretical techniques for precise control and modelling of complex quantum computers. In particular, we present a simple, parameter-free but predictive model of decoherence effects14 in our system.

Your institute does not have access to this article

Relevant articles

Open Access articles citing this article.

Access options

Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Figure 1: Quantum circuit for Shor's algorithm.
Figure 2: Structure and properties of the quantum computer molecule, a perfluorobutadienyl iron complex with the inner two carbons 13C-labelled.
Figure 3: NMR spectra at different stages in the computation.
Figure 4: Pulse sequence for implementation of the quantum circuit of Fig. 1 for a = 7.


  1. Knuth, D. E. The Art of Computer Programming Vol. 2, Seminumerical Algorithms (Addison-Wesley, Reading, Massachusetts, 1998).

    MATH  Google Scholar 

  2. Koblitz, N. A Course in Number Theory and Cryptography (Springer, New York, 1994).

    Book  Google Scholar 

  3. Bennett, C. H. & DiVincenzo, D. P. Quantum information and computation. Nature 404, 247–255 (2000).

    ADS  CAS  Article  Google Scholar 

  4. Shor, P. in Proc. 35th Annu. Symp. on the Foundations of Computer Science (ed. Goldwasser, S.) 124–134 (IEEE Computer Society Press, Los Alamitos, California, 1994).

    Book  Google Scholar 

  5. Shor, P. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26, 1484–1509 (1997).

    MathSciNet  Article  Google Scholar 

  6. Ekert, A. & Jozsa, R. Quantum computation and Shor's factoring algorithm. Rev. Mod. Phys. 68(3), 733–753 (1996).

    ADS  MathSciNet  Article  Google Scholar 

  7. Beckman, D., Chari, A. N., Devabhaktuni, S. & Preskill, J. Efficient networks for quantum factoring. Phys. Rev. A 54, 1034–1063 (1996).

    ADS  MathSciNet  CAS  Article  Google Scholar 

  8. Jones, J. A. NMR quantum computation. Prog. NMR Spectrosc. 38, 325–360 (2001).

    CAS  Article  Google Scholar 

  9. Vandersypen, L. M. K. et al. Experimental realization of an order-finding algorithm with an NMR quantum computer. Phys. Rev. Lett. 85, 5452–5455 (2000).

    ADS  CAS  Article  Google Scholar 

  10. Knill, E., Laflamme, R., Martinez, R. & Tseng, C.-H. An algorithmic benchmark for quantum information processing. Nature 404, 368–370 (2000).

    ADS  CAS  Article  Google Scholar 

  11. Gershenfeld, N. & Chuang, I. L. Bulk spin-resonance quantum computation. Science 275, 350–356 (1997).

    MathSciNet  CAS  Article  Google Scholar 

  12. Cory, D. G., Fahmy, A. F. & Havel, T. F. Ensemble quantum computing by NMR spectroscopy. Proc. Natl Acad. Sci. 94, 1634–1639 (1997).

    ADS  CAS  Article  Google Scholar 

  13. Schulman, L. & Vazirani, U. in Proc. 31st ACM Symp. on Theory of Computing 322–329 (Association for Computing Machinery, New York, 1999).

    Google Scholar 

  14. Chuang, I. L., Laflamme, R., Shor, P. & Zurek, W. H. Quantum computers, factoring, and decoherence. Science 270, 1633–1635 (1995).

    ADS  MathSciNet  CAS  Article  Google Scholar 

  15. Freeman, R. Spin Choreography (Spektrum, Oxford, 1997).

    Google Scholar 

  16. Braunstein, S. L. et al. Separability of very noisy mixed states and implications for NMR quantum computing. Phys. Rev. Lett. 83, 1054–1057 (1999).

    ADS  CAS  Article  Google Scholar 

  17. Schack, R. & Caves, C. M. Classical model for bulk-ensemble NMR quantum computation. Phys. Rev. A 60, 4354–4362 (1999).

    ADS  CAS  Article  Google Scholar 

  18. Linden, N. & Popescu, S. Good dynamics versus bad kinematics: Is entanglement needed for quantum computation? Phys. Rev. Lett. 87, 047901 (2001).

    ADS  CAS  Article  Google Scholar 

  19. Coppersmith, D. An Approximate Fourier Transform Useful in Quantum Factoring (IBM Res. Rep. RC19642, IBM T. J. Watson Research Centre, Yorktown Heights, New York, 1994).

    Google Scholar 

  20. Vold, R. L. & Vold, R. R. Nuclear magnetic relaxation in coupled spin systems. Prog. NMR Spectrosc. 12, 79–133 (1978).

    CAS  Article  Google Scholar 

  21. Jeener, J. Superoperators in magnetic resonance. Adv. Magn. Reson. 10, 1–51 (1982).

    CAS  Article  Google Scholar 

  22. Bloch, F. Nuclear induction. Phys. Rev. 70, 460–474 (1946).

    ADS  CAS  Article  Google Scholar 

  23. Kraus, K. States, Effects, and Operations: Fundamental Notions of Quantum Theory (Springer, Berlin, 1983).

    Book  Google Scholar 

  24. Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge Univ. Press, Cambridge, 2000).

    MATH  Google Scholar 

  25. Kane, B. E. A silicon-based nuclear spin quantum computer. Nature 393, 133–137 (1998).

    ADS  CAS  Article  Google Scholar 

  26. Loss, D. & DiVincenzo, D. P. Quantum computation with quantum dots. Phys. Rev. A 57, 120–126 (1998).

    ADS  CAS  Article  Google Scholar 

  27. Vandersypen, L. M. K. et al. Implementation of a three-quantum-bit search algorithm. Appl. Phys. Lett. 76, 646–648 (2000).

    ADS  CAS  Article  Google Scholar 

  28. Aho, A. V., Sethi, R. & Ullman, J. D. Compilers: Principles, Techniques and Tools (Addison-Wesley, Reading, Massachusetts, 1986).

    MATH  Google Scholar 

  29. Green, M., Mayne, N. & Stone, F. G. A. Chemistry of the metal carbonyls. Part XLVI. Perfluorobutadienyl iron, rhenium and manganese complexes. J. Chem. Soc. A 902–905 (1968).

  30. Leung, D. W. et al. Efficient implementation of selective recoupling in heteronuclear spin systems using Hadamard matrices. Phys. Rev. A 61, 042310 (2000).

    ADS  Article  Google Scholar 

Download references


We thank X. Zhou and J. Preskill for discussions, J. Smolin for the use of his IBM workstation, D. Miller for help with spectral analysis, A. Schwartz and his team for their technical assistance, and J. Harris, W. Risk and H. Coufal for their support. L.V. acknowledges a Yansouni Family Stanford graduate fellowship. This work was supported in part by the QuARC project under a DARPA Quantum Information Science and Technology grant.

Author information

Authors and Affiliations


Rights and permissions

Reprints and Permissions

About this article

Cite this article

Vandersypen, L., Steffen, M., Breyta, G. et al. Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance. Nature 414, 883–887 (2001).

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI:

Further reading


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing