Quantum Information Processing

, Volume 11, Issue 3, pp 787–839 | Cite as

Tools in the Riemannian geometry of quantum computation

Article
First Online:
Received:
Accepted:
  • 224 Downloads

Abstract

An introduction is first given of recent developments in the Riemannian geometry of quantum computation in which the quantum evolution is represented in the tangent space manifold of the special unitary unimodular group for n qubits. The Riemannian right-invariant metric, connection, curvature, geodesic equation for minimal complexity quantum circuits, Jacobi equation, and the lifted Jacobi equation for varying penalty parameter are reviewed. Sharpened tools for calculating the geodesic derivative are presented. The geodesic derivative may facilitate the numerical investigation of conjugate points and the global characteristics of geodesic paths in the group manifold, the determination of optimal quantum circuits for carrying out a quantum computation, and the determination of the complexity of particular quantum algorithms.

Keywords

Quantum computing Quantum circuits Quantum complexity Differential geometry Riemannian geometry Geodesics Lax equation Jacobi fields 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Nielsen M.A., Chuang I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  2. 2.
    Montgomery R.: A Tour of Sub-Riemannian Geometries, Their Geodesics and Applications, Vol. 91 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2002)Google Scholar
  3. 3.
    Khaneja N., Glaser S.J., Brockett R.: Sub-Riemannian geometry and time optimal control of three spin systems: quantum gates and coherence transfer. Phys. Rev. A 65(1–11), 032301 (2002)MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Moseley C.G.: Geometric control of quantum spin systems. In: Donkor, E., Pirich, A.R., Brandt, H.E. (eds) Quantum Information and Computation II, Proceedings of the SPIE vol. 5436., pp. 319–323. SPIE, Bellingham (2004)Google Scholar
  5. 5.
    Nielsen M.A.: A geometric approach to quantum circuit lower bounds. Quantum Inf. Comput. 6, 213–262 (2006)MathSciNetMATHGoogle Scholar
  6. 6.
    Nielsen M.A., Dowling M.R., Gu M., Doherty A.C.: Optimal control, geometry, and quantum computing. Phys. Rev. A 73(1–7), 062323 (2006)MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Nielsen M.A., Dowling M.R., Gu M., Doherty A.C.: Quantum computation as geometry. Science 311, 1133–1135 (2006)MathSciNetADSMATHCrossRefGoogle Scholar
  8. 8.
    Dowling M.R., Nielsen M.A.: The geometry of quantum computation. Quantum Inf. Comput. 8, 0861–0899 (2008)MathSciNetGoogle Scholar
  9. 9.
    Clelland J.N., Moseley C.G.: Sub-Finsler geometry in dimension three. Differ. Geome. Appl. 24, 628–651 (2006)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Hou B.-Y., Hou B.-Y.: Differential Geometry for Physicists. World Scientific, Singapore (1997)MATHCrossRefGoogle Scholar
  11. 11.
    Sagle A.A., Walde R.E.: Introduction to Lie Groups and Lie Algebras. Academic Press, New York (1973)MATHGoogle Scholar
  12. 12.
    Conlon L.: Differentiable Manifolds, 2nd edn. Birkhäuser, Boston (2001)MATHCrossRefGoogle Scholar
  13. 13.
    Lee J.M.: Riemannian Manifolds: An Introduction to Curvature. Springer, New York (1997)MATHGoogle Scholar
  14. 14.
    Berger M.: A Panoramic View of Riemannian Geometry. Springer, Berlin (2003)MATHCrossRefGoogle Scholar
  15. 15.
    Milnor J.: Morse Theory. Princeton University Press, Princeton (1973)Google Scholar
  16. 16.
    Hall B.C.: Lie Groups, Lie Algebras, and Representations. Springer, New York (2004)Google Scholar
  17. 17.
    Farout J.: Analysis on Lie Groups. Cambridge University Press, Cambridge (2008)CrossRefGoogle Scholar
  18. 18.
    Postnikov M.M.: Geometry VI: Riemannian Geometry, Encyclopedia of Mathematical Sciences, vol. 91. Springer, Berlin (2001)Google Scholar
  19. 19.
    Petersen P.: Riemannian Geometry, 2nd edn. Springer, New York (2006)MATHGoogle Scholar
  20. 20.
    Jost J.: Riemannian Geometry and Geometric Analysis, 5th edn. Springer, Berlin (2008)MATHGoogle Scholar
  21. 21.
    Wasserman R.: Tensors and Manifolds, 2nd edn. Oxford University Press, Oxford (2004)MATHGoogle Scholar
  22. 22.
    Naimark M.A., Stern A.I.: Theory of Group Representations. Springer, New York (1982)MATHCrossRefGoogle Scholar
  23. 23.
    Sepanski M.R.: Compact Lie Groups. Springer, New York (2007)MATHCrossRefGoogle Scholar
  24. 24.
    Pfeifer W.: The Lie Algebras su(N). Birkhäuser, Basel (2003)MATHCrossRefGoogle Scholar
  25. 25.
    Stillwell J.: Naive Lie Theory. Springer, NY (2008)MATHCrossRefGoogle Scholar
  26. 26.
    Cornwell J.F.: Group Theory in Physics: An Introduction. Academic Press, San Diego (1997)MATHGoogle Scholar
  27. 27.
    Cornwell J.F.: Group Theory in Physics, vols. 1 & 2. Academic Press, London (1984)Google Scholar
  28. 28.
    Steeb W., Hardy Y.: Problems and Solutions in Quantum Computing and Quantum Information, 2nd edn. World Scientific, New Jersey (2006)MATHGoogle Scholar
  29. 29.
    Weigert S.: Baker-Campbell-Hausdorff relation for special unitary groups SU(N). J. Phys. A Math. Gen. 30, 8739–8749 (1997)MathSciNetADSMATHCrossRefGoogle Scholar
  30. 30.
    Reutenauer C.: Free Lie Algebras. Clarendon Press, Oxford (1993)MATHGoogle Scholar
  31. 31.
    Dynkin E.: Calculation of the coefficients in the Campbell-Hausdorff formula. Dokl. Akad. Nauk 57, 323–326 (1947)MathSciNetMATHGoogle Scholar
  32. 32.
    Baker, H.F.: Alternants and continuous groups. In: Proceedings of the London Mathematical Society, vol. 3, 2nd edn., pp. 24–47 (1905)Google Scholar
  33. 33.
    Campbell, J.E. On a law of combination of operators bearing on the theory of continuous transformation groups. In: Proceedings of the London Mathematical Society, vol. 28, 1st edn., pp. 381–390 (1897)Google Scholar
  34. 34.
    Campbell, J.E.: On a law of combination of operators. In: Proceedings of the London Mathematical Society, vol. 29, 1st edn., pp. 14–32 (1898)Google Scholar
  35. 35.
    Hausdorff F.: Die symbolische Exponentialformel in der Gruppentheorie. Leipziger Berichte 58, 19–48 (1906)Google Scholar
  36. 36.
    Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation, pp. 223, 224, 310. W. H. Freeman and Company, New York (1973)Google Scholar
  37. 37.
    Lax P.D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21, 467–490 (1968)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Abraham R., Marsden J.E.: Foundations of Mechanics, 2nd edn. AMS Chelsea Publishing, American Mathematical Society, Providence (2008)Google Scholar
  39. 39.
    Zwillinger D.: Handbook of Differential Equations, 3rd edn. Academic Press, San Diego (1998)MATHGoogle Scholar
  40. 40.
    Kaushal R.S., Parashar D.: Advanced Methods of Mathematical Physics. CRC Press, Boca Raton (2000)Google Scholar
  41. 41.
    Miwa T., Jimbo M., Date E.: Solitons. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  42. 42.
    Debnath L.: Nonlinear Partial Differential Equations. Birkhäuser, Boston (1997)MATHGoogle Scholar
  43. 43.
    Zeidler E.: Nonlinear Functional Analysis and its Applications IV: Applications to Mathematical Physics. Springer, New York (1997)Google Scholar
  44. 44.
    Brandt, H.E.: Riemannian geometry of quantum computation. In: Lomonaco, S.J. (ed.) Proceedings of the Symposia Applied Mathematics, vol. 68, pp. 61–101 (2010)Google Scholar
  45. 45.
    Milnor J.: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21, 293–329 (1976)MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Arnold V.I.: Mathematical Methods of Classical Mechanics, 2nd edn. Springer, New York (1989)Google Scholar
  47. 47.
    Arnold V.I., Khesin B.A.: Topological Methods in Hydrodynamics. Springer, New York (1999)Google Scholar
  48. 48.
    Brandt H.E.: Riemannian curvature in the differential geometry of quantum computation. Phys. E 42, 449–453 (2010)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Brandt H.E.: Riemannian geometry of quantum computation. Nonlinear Anal. 71, e474–e485 (2009)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Milnor J.: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21, 293–329 (1976)MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Greiner W., Reinhardt J.: Field Quantization, vol. 219. Springer, Berlin (1996)Google Scholar
  52. 52.
    Weinberg S.: The Quantum Theory of Fields I, vol. 143. Cambridge University Press, Cambridge (1995)Google Scholar
  53. 53.
    Horn R., Johnson C.: Topics in Matrix Analysis, vol. 255. Cambridge University Press, Cambridge (1991)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC (outside the USA)  2011

Authors and Affiliations

  1. 1.US Army Research LaboratoryAdelphiUSA

Personalised recommendations