Systematic analysis of majorization in quantum algorithms

  • R. Orús
  • J. I. Latorre
  • M. A. Martín-Delgado


Motivated by the need to uncover some underlying mathematical structure of optimal quantum computation, we carry out a systematic analysis of a wide variety of quantum algorithms from the majorization theory point of view. We conclude that step-by-step majorization is found in the known instances of fast and efficient algorithms, namely in the quantum Fourier transform, in Grover’s algorithm, in the hidden affine function problem, in searching by quantum adiabatic evolution and in deterministic quantum walks in continuous time solving a classically hard problem. On the other hand, the optimal quantum algorithm for parity determination, which does not provide any computational speed-up, does not show step-by-step majorization. Lack of both speed-up and step-by-step majorization is also a feature of the adiabatic quantum algorithm solving the 2-SAT “ring of agrees” problem. Furthermore, the quantum algorithm for the hidden affine function problem does not make use of any entanglement while it does obey majorization. All the above results give support to a step-by-step Majorization Principle necessary for optimal quantum computation.


Fourier Fourier Transform Continuous Time Efficient Algorithm Theory Point 
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. 1.
    L.K. Grover, Phys. Rev. Lett. 78, 325 (1997)CrossRefGoogle Scholar
  2. 2.
    P.W. Shor, Proc. 35th IEEE (IEEE press, Los Alamitos CA, 1994), p. 352; quant-ph/9508027Google Scholar
  3. 3.
    E. Farhi, J. Goldstone, S. Gutmann, M. Sipser, quant-ph/0001106Google Scholar
  4. 4.
    A.M. Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann, D. Spielman, quant-ph/0209131Google Scholar
  5. 5.
    J. Ahn, T.C. Weinacht, P.H. Bucksbaum, Science 287, 463 (2000)CrossRefGoogle Scholar
  6. 6.
    P. Knight, Science 287, 441 (2000)CrossRefGoogle Scholar
  7. 7.
    S. Lloyd, quant-ph/9903057Google Scholar
  8. 8.
    R. Jozsa, N. Linden, quant-ph/0201143Google Scholar
  9. 9.
    A. Galindo, M.A. Martí n-Delgado, Rev. Mod. Phys. 74, 347 (2002); quant-ph/0112105CrossRefGoogle Scholar
  10. 10.
    G. Vidal, quant-ph/0301063Google Scholar
  11. 11.
    R.F. Muirhead, Proc. Edinburg Math. Soc. 21, 144 (1903)Google Scholar
  12. 12.
    G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities (Cambridge University Press, 1978)Google Scholar
  13. 13.
    A.W. Marshall, I. Olkin, Inequalities: Theory of Majorization and its Applications (Acad. Press Inc., 1979)Google Scholar
  14. 14.
    R. Bathia, Matrix Analysis, Graduate Texts in Mathematics (Springer-Verlag, 1996), Vol. 169Google Scholar
  15. 15.
    M.A. Nielsen, G. Vidal, Quant. Inform. Comput. 1, 76 (2001)Google Scholar
  16. 16.
    J.I. Latorre, M.A. Martín-Delgado, Phys. Rev. A 66, 022305 (2002); quant-ph/0111146CrossRefGoogle Scholar
  17. 17.
    R. Orús, J.I. Latorre, M.A. Martín-Delgado, Quant. Inform. Proc. 4, 283 (2003); quant-ph/0206134Google Scholar
  18. 18.
    E. Bernstein, U. Vazirani, Quant. Compl. Theor. SIAM J. Comp. 26(5), 1411 (1997)zbMATHGoogle Scholar
  19. 19.
    D. Deutsch, Proc. R. Soc. Lond. A 400, 97 (1985)zbMATHGoogle Scholar
  20. 20.
    R. Cleve, A. Ekert, C. Macchiavello, M. Mosca, Proc. R. Soc. Lond. A 454, 339 (1998)CrossRefzbMATHGoogle Scholar
  21. 21.
    M. Mosca, Ph.D. thesis, 1999Google Scholar
  22. 22.
    M.A. Nielsen, I. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000)Google Scholar
  23. 23.
    D. Coppersmith, IBM Research Report Report 19642, 1994; quant-ph/0201067Google Scholar
  24. 24.
    E. Farhi, J. Goldstone, S. Gutmann, M. Sipser, Phys. Rev. Lett. 81, 5442 (1998); quant-ph/9802045CrossRefGoogle Scholar
  25. 25.
    R. Beals, H. Buhrman, R. Cleve, M. Mosca, R. de Wolf, Proc. of the 99th Annual Symposium on Foundations of Computer Science (FOCS’98) (IEEE press, Los Alamitos CA, 1998), pp. 352-361; quant-ph/9802049Google Scholar
  26. 26.
    S. Das, R. Kobes, G. Kunstatter, quant-ph/0204044Google Scholar
  27. 27.
    J. Roland, N.J. Cerf, Phys. Rev. A 65, 042308 (2002); quant-ph/0107015CrossRefGoogle Scholar
  28. 28.
    W. van Dam, M. Mosca, U. Vazirani, Proceedings of the 42nd Annual Symposium of Computer Science (IEEE press, 2001), pp. 279-287; quant-ph/0206003Google Scholar
  29. 29.
    J. Roland, N.J. Cerf, quant-ph/0302138Google Scholar
  30. 30.
    S. A. Cook, Proc. 3rd Ann. ACM Symp. on Theory of Computing (Association for Computing Machinery, New York, 1971), pp. 151-158Google Scholar
  31. 31.
    Y. Aharonov, L. Davidovich, N. Zagury, Phys. Rev. A 48, 1687 (1993)Google Scholar
  32. 32.
    D. Aharonov, A. Ambainis, J. Kempe, U. Vazirani, Proc. of the 33rd ACM Symposium on the Theory of Computing (ACM Press, New York, 2001), p. 50Google Scholar
  33. 33.
    A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, J. Watrous, Proc. 33rd Symposium on the Theory of Computing (ACM Press, New York, 2001), p. 37Google Scholar
  34. 34.
    E. Farhi, S. Gutmann, Phys. Rev. A 58, 915 (1998); quant-ph/9706062CrossRefMathSciNetGoogle Scholar
  35. 35.
    A.M. Childs, E. Farhi, S. Gutmann, Quant. Inform. Proc. 1, 35 (2002); quant-ph/0103020CrossRefGoogle Scholar
  36. 36.
    J. Kempe, quant-ph/0205083Google Scholar
  37. 37.
    N. Shenvi, J. Kempe, K. Birgitta Whaley, quant-ph/0210064Google Scholar

Copyright information

© Springer-Verlag Berlin/Heidelberg 2004

Authors and Affiliations

  • R. Orús
    • 1
  • J. I. Latorre
    • 1
  • M. A. Martín-Delgado
    • 2
  1. 1.Dept. d’Estructura i Constituents de la MatériaUniv. BarcelonaBarcelonaSpain
  2. 2.Departamento de Física Teórica IUniversidad ComplutenseMadridSpain

Personalised recommendations