Skip to main content
SpringerLink
Log in

Zero entries distribution in a unitary matrix

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

We present a principle to determine whether a matrix is unitary. Through this principle, we can exclude some zero entries distribution in unitary matrices. As a result, we can construct higher order unitary matrices with a given zero entries distribution. We apply our results to construct CP-equivalent gates.

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

Similar content being viewed by others

Data Availability

All data supporting the findings of this study are available within the article or from the corresponding author upon reasonable request.

References

  1. De Baerdemacker, S., De Vos, A., Chen, L., Li, Yu.: The Birkhoff theorem of unitary matrices of arbitrary dimensions. Linear Algebra Appl. 514, 151–164 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Muller-Hermes, A., Nechita, I.: Operator Schmidt ranks of bipartite unitary matrices. Linear Algebra Appl. 557, 174–187 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ouyang, Y., Shen, Y., Chen, L.: Faster quantum computation with permutations and resonant couplings. Linear Algebra Appl. 592, 270–286 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  4. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  5. Benenti, G., Casati, G.: Giuliano Strini. World Scientific, Principles of Quantum Computation and Information (2004)

    Google Scholar 

  6. Kliuchnikov, V., Maslov, D., Mosca, M.: Asymptotically optimal approximation of single qubit Unitaries by Clifford and t-circuits using a constant number of ancillary qubits. Phys. Rev. Lett. 110, 190502 (2013)

    Article  ADS  Google Scholar 

  7. Malinovsky, V.S., Sola, I.R., Vala, J.: Phase-controlled two-qubit quantum gates. Phys. Rev. A 89, 032301 (2014)

    Article  ADS  Google Scholar 

  8. Bennett, C.H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein–Podolsk–Rosen channels. Phys. Rev. Lett. 70, 1895–1899 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Liang, M., Mengyao, H., Sun, Y., Chen, L., Chen, X.: Real entries of complex Hadamard matrices and mutually unbiased bases in dimension six. Linear and Multilinear Algebra 69(15), 1–18 (2019)

    MathSciNet  MATH  Google Scholar 

  10. Szöllősi, F.: Parametrizing complex Hadamard matrices. Eur. J. Combinat. 29(5), 1219–1234 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Werner, R.F.: All teleportation and dense coding schemes. J. Phys. A 34, 7081–7094 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Chen, L., Li, Y.: Nonlocal and controlled unitary operators of Schmidt rank three. Phys. Rev. A 89, 062326 (2014)

    Article  ADS  Google Scholar 

  13. Shende, V., Markov, I., Bullock, S.: Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69(6), 666–670 (2003)

    Google Scholar 

  14. Kempe, J.: Multiparticle entanglement and its applications to crytography. Phys. Rev. A 60(2), 910–916 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  15. Nielsen, M.A., Dawson, C.M., Dodd, J.L., Gilchrist, A., Mortimer, D., Osborne, T.J., Bremner, M.J., Harrow, A.W., Hines, A.: Quantum dynamics as a physical resource. Phys. Rev. A 67(5), 052301 (2003)

    Article  ADS  Google Scholar 

  16. Hein, M., Eisert, J., Briegel, H.J.: Multiparty entanglement in graph states. Phys. Rev. A 69, 062311 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Barreiro, J.T., Schindler, P., Gühne, O., Monz, T., Chwalla, M., Roos, C.F., Hennrich, M., Blatt, R.: Experimental multiparticle entanglement dynamics induced by decoherence. Nat. Phys. 6, 943–946 (2010)

    Article  Google Scholar 

  18. Feng, C., Chen, L.: Zero entries in multipartite product unitary matrices. Q. Inf. Process. 20, 250 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  19. Tillmann, M., Dakic, B., Heilmann, R., Nolte, S., Szameit, A., Walther, P.: Experimental boson sampling. Nat. Photonics 7(7), 540–544 (2013)

    Article  ADS  Google Scholar 

  20. Reck, M., Zeilinger, A., Bernstein, H., Bertani, P.: Experimental realization of any discrete unitary operator. Phys. Rev. Lett. 73(1), 58–61 (1994)

    Article  ADS  Google Scholar 

  21. Li, C.-K., Roberts, R., Yin, X.: Decomposition of unitary matrices and quantum gates. Int. J. Q. Inf. 11(01), 1350015 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Song, Z.W., Chen, L.: On the zero entries in a unitary matrix. Linear Multilinear Algebra 70(7), 1–10 (2020)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

Authors thank the interesting discussion with Professor Chuanyu Wang and Wei Wang. CCF and LC were supported by the NNSF of China (Grant No. 11871089), and the Fundamental Research Funds for the Central Universities(Grant No. ZG216S2005).

Author information

Authors and Affiliations

  1. School of Mathematics, Physics and Finance, Anhui Polytechnic University, Wuhu, 241000, China

    Shaomin Liu

  2. LMIB(Beihang University), Ministry of Education, and School of Mathematical Sciences, Beihang University, Beijing, 100191, China

    Changchun Feng & Lin Chen

  3. International Research Institute for Multidisciplinary Science, Beihang University, Beijing, 100191, China

    Lin Chen

Authors
  1. Shaomin Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Changchun Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lin Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Changchun Feng or Lin Chen.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest to this work.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, S., Feng, C. & Chen, L. Zero entries distribution in a unitary matrix. Quantum Inf Process 22, 169 (2023). https://doi.org/10.1007/s11128-023-03913-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-023-03913-6

Keywords

Search

Navigation