Skip to main content
SpringerLink
Log in

On sets of maximally commuting and anticommuting Pauli operators

  • Research
  • Published:
Research in the Mathematical Sciences Aims and scope Submit manuscript

Abstract

In this work, we study the structure and cardinality of maximal sets of commuting and anticommuting Paulis in the setting of the abelian Pauli group. We provide necessary and sufficient conditions for anticommuting sets to be maximal and present an efficient algorithm for generating anticommuting sets of maximum size. As a theoretical tool, we introduce commutativity maps and study properties of maps associated with elements in the cosets with respect to anticommuting minimal generating sets. We also derive expressions for the number of distinct sets of commuting and anticommuting abelian Paulis of a given size.

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

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Alon, N., Lubetzky, E.: Codes and Xor graph products. Combinatorica 27(1), 13–33 (2007)

    Article  MathSciNet  Google Scholar 

  2. Alon, N., Lubetzky, E.: Graph powers, Delsarte, Hoffman, Ramsey, and Shannon. SIAM J. Discrete Math. 21(2), 329–348 (2007)

    Article  MathSciNet  Google Scholar 

  3. Bonet-Monroig, X., Babbush, R., O’Brien, T.E.: Nearly optimal measurement scheduling for partial tomography of quantum states. arXiv preprint arXiv:1908.05628 (2019)

  4. Bravyi, S.B., Kitaev, A.Y.: Fermionic quantum computation. Ann. Phys. 298(1), 210–226 (2002)

    Article  MathSciNet  Google Scholar 

  5. Calderbank, A., Naguib, A.: Orthogonal designs and third generation wireless communication. London Mathematical Society Lecture Note Series pp. 75–107 (2001)

  6. Gottesman, D.E.: Stabilizer codes and quantum error correction. dissertation. Ph.D. thesis, California Institute of Technology (1997). http://resolver.caltech.edu/CaltechETD:etd-07162004-113028

  7. Hoffman, D.G., Leonard, D.A., Lindner, C.C., Phelps, K., Rodger, C., Wall, J.R.: Coding Theory: The Essentials. Mercel Dekker, New York (1992)

    MATH  Google Scholar 

  8. Hrubeš, P.: On families of anticommuting matrices. Linear Algebra Appl. 493, 494–507 (2016)

    Article  MathSciNet  Google Scholar 

  9. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  10. Raussendorf, R., Bermejo-Vega, J., Tyhurst, E., Okay, C., Zurel, M.: Phase-space-simulation method for quantum computation with magic states on qubits. Phys. Rev. A 101, 012350 (2020). https://doi.org/10.1103/PhysRevA.101.012350

  11. Rotman, J.J.: An introduction to the theory of groups, 4th edn. Graduate texts in mathematics. Springer (1999)

  12. Saniga, M., Planat, M.: Multiple qubits as symplectic polar spaces of order two. Adv. Stud. Theor. Phys. 1(1), 1–4 (2007)

    MathSciNet  MATH  Google Scholar 

  13. Sloane, N.J.: The on-line encyclopedia of integer sequences. Sequence A128036, https://oeis.org/A128036

  14. Wigner, E.P., Jordan, P.: Über das paulische äquivalenzverbot. Z. Phys 47, 631 (1928)

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous reviewer whose suggestions helped to greatly improve the paper. The authors would also like to thank Sergey Bravyi, Kristan Temme, and Ted Yoder for useful discussions. R.S. would like to thank IBM T.J. Watson Research Center for facilitating the research.

Author information

Authors and Affiliations

  1. Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA, USA

    Rahul Sarkar

  2. IBM T.J. Watson Research Center, Yorktown Heights, NY, USA

    Ewout van den Berg

Authors
  1. Rahul Sarkar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ewout van den Berg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ewout van den Berg.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sarkar, R., van den Berg, E. On sets of maximally commuting and anticommuting Pauli operators. Res Math Sci 8, 14 (2021). https://doi.org/10.1007/s40687-020-00244-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40687-020-00244-1

Keywords

Search

Navigation