Designs, Codes and Cryptography

, Volume 53, Issue 1, pp 13–31 | Cite as

Unitary designs and codes

  • Aidan Roy
  • A. J. Scott


A unitary design is a collection of unitary matrices that approximate the entire unitary group, much like a spherical design approximates the entire unit sphere. In this paper, we use irreducible representations of the unitary group to find a general lower bound on the size of a unitary t-design in U(d), for any d and t. We also introduce the notion of a unitary code—a subset of U(d) in which the trace inner product of any pair of matrices is restricted to only a small number of distinct absolute values—and give an upper bound for the size of a code with s inner product values in U(d), for any d and s. These bounds can be strengthened when the particular inner product values that occur in the code or design are known. Finally, we describe some constructions of designs: we give an upper bound on the size of the smallest weighted unitary t-design in U(d), and we catalogue some t-designs that arise from finite groups.


Unitary design Unitary code Unitary group Rank bound Linear programming bound Delsarte bound Spherical design Spherical code Quantum process tomography Zonal polynomial 

Mathematics Subject Classification (2000)

05B30 41A55 81P15 94A20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Dankert C.: Efficient simulation of random quantum states and operators. Master’s thesis, University of Waterloo (2005).
  2. 2.
    Dankert C., Cleve R., Emerson J., Livine E.: Exact and approximate unitary 2-designs: constructions and applications (2006).
  3. 3.
    Gross D., Audenaert K., Eisert J.: Evenly distributed unitaries: on the structure of unitary designs. J. Math. Phys. 48(5), 052104, 22 pp. (2007).Google Scholar
  4. 4.
    Hayden P., Preskill J.: Black holes as mirrors: quantum information in random subsystems. J. High Energy Phys. 2007(9), 120, 21 pp. (2007).Google Scholar
  5. 5.
    Scott A.J.: Optimizing quantum process tomography with unitary 2-designs. J. Phys. A 41(5), 055308, 26 pp. (2008).Google Scholar
  6. 6.
    Harrow A., Low R.: Random quantum circuits are approximate 2-designs. Comm. Math. Phys. (2008) (accepted).
  7. 7.
    Ambainis A., Bouda J., Winter A.: Non-malleable encryption of quantum information. J. Math. Phys. (2008) (accepted).
  8. 8.
    Harrow A., Low R.: Efficient quantum tensor product expanders and k-designs. (2008).
  9. 9.
    Collins B.: Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability. Int. Math. Res. Not. 17, 953–982 (2003)CrossRefGoogle Scholar
  10. 10.
    Collins B., Śniady P.: Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Comm. Math. Phys. 264(3), 773–795 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Welch L.: Lower bounds on the maximum cross correlation of signals. IEEE Trans. Inform. Theory 20(3), 397–399 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Diaconis P., Shahshahani M.: On the eigenvalues of random matrices. J. Appl. Probab. 31A, 49–62 (1994)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Rains E.M.: Increasing subsequences and the classical groups. Electron. J. Comb. 5, 9 pp. (1998) (Research Paper 12).Google Scholar
  14. 14.
    Hayashi A., Hashimoto T., Horibe M.: Reexamination of optimal quantum state estimation of pure states. Phys. Rev. A 72(3), 032325, 5 pp. (2005).Google Scholar
  15. 15.
    Ambainis A., Emerson J.: Quantum t-designs: t-wise independence in the quantum world. In: Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity, pp. 129–140 (2007).Google Scholar
  16. 16.
    Delsarte P., Goethals J.M., Seidel J.J.: Bounds for systems of lines, and Jacobi polynomials. Philips Res. Rep. 30, 91–105 (1975)zbMATHGoogle Scholar
  17. 17.
    Delsarte P., Goethals J.M., Seidel J.J.: Spherical codes and designs. Geometriae Dedicata 6(3), 363–388 (1977)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Bump D.: Graduate Texts in Mathematics. Lie Groups, vol. 225. Springer, New York (2004)Google Scholar
  19. 19.
    Sepanski M.R.: Graduate Texts in Mathematics. Compact Lie Groups, vol. 235. Springer, New York (2007)Google Scholar
  20. 20.
    Stembridge J.R.: Rational tableaux and the tensor algebra of gl n. J. Comb. Theory Ser. A 46(1), 79–120 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Stembridge J.R.: A combinatorial theory for rational actions of GL n. In: Invariant Theory, Contemporary Mathematics, vol. 88, pp. 163–176. American Mathematical Society, Providence, RI (1989).Google Scholar
  22. 22.
    Benkart G., Chakrabarti M., Halverson T., Leduc R., Lee C., Stroomer J.: Tensor product representations of general linear groups and their connections with Brauer algebras. J. Algebra 166(3), 529–567 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Godsil C.D.: Polynomial spaces. Discret. Math. 73(1–2), 71–88 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Godsil C.D.: Algebraic Combinatorics. Chapman & Hall, New York (1993)zbMATHGoogle Scholar
  25. 25.
    Delsarte P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. 10(Suppl), vi+97 (1973)MathSciNetGoogle Scholar
  26. 26.
    Levenshtein V.: On designs in compact metric spaces and a universal bound on their size. Discret. Math. 192(1–3), 251–271 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Neumaier A.: Combinatorial configurations in terms of distances. Eindhoven University of Technology Memorandum (81–09) (1981).Google Scholar
  28. 28.
    Hoggar S.G.: t-designs in Delsarte spaces. In: Coding Theory and Design Theory, Part II. Institute for Mathematics and its Application Volume Series, vol. 21, pp. 144–165. Springer, New York (1990).Google Scholar
  29. 29.
    Bachoc C., Coulangeon R., Nebe G.: Designs in Grassmannian spaces and lattices. J. Algebraic Comb. 16(1), 5–19 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Bachoc C., Bannai E., Coulangeon R.: Codes and designs in Grassmannian spaces. Discret. Math. 277(1–3), 15–28 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Bachoc C.: Linear programming bounds for codes in Grassmannian spaces. IEEE Trans. Inform. Theory 52(5), 2111–2125 (2006)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Roy A.: Bounds for codes and designs in complex subspaces. J. Algebraic Comb. (2008) (accepted).
  33. 33.
    Renes J., Blume-Kohout R., Scott A.J., Caves C.M.: Symmetric informationally complete quantum measurements. J. Math. Phys. 45, 2171 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Klappenecker A., Rötteler M.: Unitary error bases: constructions, equivalence, and applications. In: Applied Algebra, Algebraic Algorithms and Error-correcting Codes. Lecture Notes in Computer Science, vol. 2643, pp. 139–149. Springer, Berlin (2003).Google Scholar
  35. 35.
    de la Harpe P., Pache C.: Cubature formulas, geometrical designs, reproducing kernels, and Markov operators. In: Infinite Groups: Geometric, Combinatorial and Dynamical Aspects. Progress in Mathematics, vol. 248, pp. 219–267. Birkhäuser, Basel (2005).Google Scholar
  36. 36.
    Seymour P.D., Zaslavsky T.: Averaging sets: a generalization of mean values and spherical designs. Adv. Math. 52(3), 213–240 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Chau H.F.: Unconditionally secure key distribution in higher dimensions by depolarization. IEEE Trans. Inform. Theory 51(4), 1451–1468 (2005)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, Institute for Quantum Information ScienceUniversity of CalgaryCalgaryCanada
  2. 2.Centre for Quantum Dynamics, Centre for Quantum Computer TechnologyGriffith UniversityBrisbaneAustralia

Personalised recommendations