Skip to main content
SpringerLink
Log in

Unitary designs and codes

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Dankert C.: Efficient simulation of random quantum states and operators. Master’s thesis, University of Waterloo (2005). http://arxiv.org/abs/quant-ph/0512217.

  2. Dankert C., Cleve R., Emerson J., Livine E.: Exact and approximate unitary 2-designs: constructions and applications (2006). http://arxiv.org/quant-ph/0606161.

  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. Hayden P., Preskill J.: Black holes as mirrors: quantum information in random subsystems. J. High Energy Phys. 2007(9), 120, 21 pp. (2007).

  5. Scott A.J.: Optimizing quantum process tomography with unitary 2-designs. J. Phys. A 41(5), 055308, 26 pp. (2008).

  6. Harrow A., Low R.: Random quantum circuits are approximate 2-designs. Comm. Math. Phys. http://arxiv.org/abs/0802.1919 (2008) (accepted).

  7. Ambainis A., Bouda J., Winter A.: Non-malleable encryption of quantum information. J. Math. Phys. http://www.arxiv.org/abs/0808.0353 (2008) (accepted).

  8. Harrow A., Low R.: Efficient quantum tensor product expanders and k-designs. http://arxiv.org/abs/0811.2597 (2008).

  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)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  11. Welch L.: Lower bounds on the maximum cross correlation of signals. IEEE Trans. Inform. Theory 20(3), 397–399 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  12. Diaconis P., Shahshahani M.: On the eigenvalues of random matrices. J. Appl. Probab. 31A, 49–62 (1994)

    Article  MathSciNet  Google Scholar 

  13. Rains E.M.: Increasing subsequences and the classical groups. Electron. J. Comb. 5, 9 pp. (1998) (Research Paper 12).

  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).

  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).

  16. Delsarte P., Goethals J.M., Seidel J.J.: Bounds for systems of lines, and Jacobi polynomials. Philips Res. Rep. 30, 91–105 (1975)

    MATH  Google Scholar 

  17. Delsarte P., Goethals J.M., Seidel J.J.: Spherical codes and designs. Geometriae Dedicata 6(3), 363–388 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  18. Bump D.: Graduate Texts in Mathematics. Lie Groups, vol. 225. Springer, New York (2004)

    Google Scholar 

  19. Sepanski M.R.: Graduate Texts in Mathematics. Compact Lie Groups, vol. 235. Springer, New York (2007)

    Google Scholar 

  20. Stembridge J.R.: Rational tableaux and the tensor algebra of gl n . J. Comb. Theory Ser. A 46(1), 79–120 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  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).

  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)

    Article  MATH  MathSciNet  Google Scholar 

  23. Godsil C.D.: Polynomial spaces. Discret. Math. 73(1–2), 71–88 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  24. Godsil C.D.: Algebraic Combinatorics. Chapman & Hall, New York (1993)

    MATH  Google Scholar 

  25. Delsarte P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. 10(Suppl), vi+97 (1973)

    MathSciNet  Google Scholar 

  26. Levenshtein V.: On designs in compact metric spaces and a universal bound on their size. Discret. Math. 192(1–3), 251–271 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  27. Neumaier A.: Combinatorial configurations in terms of distances. Eindhoven University of Technology Memorandum (81–09) (1981).

  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).

  29. Bachoc C., Coulangeon R., Nebe G.: Designs in Grassmannian spaces and lattices. J. Algebraic Comb. 16(1), 5–19 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  30. Bachoc C., Bannai E., Coulangeon R.: Codes and designs in Grassmannian spaces. Discret. Math. 277(1–3), 15–28 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  31. Bachoc C.: Linear programming bounds for codes in Grassmannian spaces. IEEE Trans. Inform. Theory 52(5), 2111–2125 (2006)

    Article  MathSciNet  Google Scholar 

  32. Roy A.: Bounds for codes and designs in complex subspaces. J. Algebraic Comb. http://arxiv.org/abs/0806.2317 (2008) (accepted).

  33. Renes J., Blume-Kohout R., Scott A.J., Caves C.M.: Symmetric informationally complete quantum measurements. J. Math. Phys. 45, 2171 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  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).

  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).

  36. Seymour P.D., Zaslavsky T.: Averaging sets: a generalization of mean values and spherical designs. Adv. Math. 52(3), 213–240 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  37. Chau H.F.: Unconditionally secure key distribution in higher dimensions by depolarization. IEEE Trans. Inform. Theory 51(4), 1451–1468 (2005)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Institute for Quantum Information Science, University of Calgary, Calgary, Alberta, T2N 1N4, Canada

    Aidan Roy

  2. Centre for Quantum Dynamics, Centre for Quantum Computer Technology, Griffith University, Brisbane, QLD, 4111, Australia

    A. J. Scott

Authors
  1. Aidan Roy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. J. Scott
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aidan Roy.

Additional information

Communicated by W. H. Haemers.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Roy, A., Scott, A.J. Unitary designs and codes. Des. Codes Cryptogr. 53, 13–31 (2009). https://doi.org/10.1007/s10623-009-9290-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-009-9290-2

Keywords

Mathematics Subject Classification (2000)

Search

Navigation