Skip to main content
SpringerLink
Log in

Correspondence between quantization schemes for two-player nonzero-sum games and CNOT complexity

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

The well-known quantization schemes for two-player nonzero-sum games are Eisert–Wilkens–Lewenstein scheme and Marinatto–Weber scheme. In this work, we establish the connection between the two schemes from the perspective of quantum circuits. Further, we provide the correspondence between any game quantization schemes and the CNOT complexity, where CNOT complexity is up to the local unitary operations. While CNOT complexity is known to be useful in the analysis of universal quantum circuit, in this work, we find its applicability in quantum game theory.

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

References

  1. Nash, J.: Equilibrium points in n- person game. Proc. Natl. Acad. Sci. 36, 48–49 (1950)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Wiley, New York (1967)

    MATH  Google Scholar 

  3. Meyer, D.A.: Quantum strategies. Phys. Rev. Lett. 82, 1052–1055 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Meyer, A.: Quantum games and quantum algorithms. In: Lomonaco Jr., S.J., Brandt, H.E. (eds.) AMS Contemporary Mathematics, Volume: Quantum Computation and Quantum Information Science, vol. 305. American Mathematical Society, Providence (2002)

    Google Scholar 

  5. Eisert, J., Wilkens, M., Lewenstein, M.: Quantum games and quantum strategies. Phys. Rev. Lett. 83, 3077–3080 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Marinatto, L., Weber, T.: A quantum approach to static games of complete information. Phys. Letts. A. 272, 291–303 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Nawaz, A., Toor, A.H.: Generalized quantization scheme for two-person non-zero sum games. J. Phys. A. 37, 11457–11464 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Iqbal, A.: Studies in the theory of quantum games. Ph.D. thesis, Quaid-I-Azam University, Islamabad, Pakistan (2004). arXiv:quant-ph/0503176

  9. Flitney, A.P.: Aspects of quantum game theory. Ph.D. thesis, University of Adelaide, Adelaide, Australia, January, 2005. https://digital.library.adelaide.edu.au/dspace/bitstream/2440/37958/10/02whole.pdf

  10. Nawaz, A.: The generalized quantization schemes for games and its application to quantum information. Ph.D. thesis, Quaid-I-Azam University, Islamabad, Pakistan (2007). arXiv:quant-ph/1012.1933

  11. Flitney, A.P., Abbott, D.: An introduction to quantum game theory. Fluct. Noise Lett. 2, R175–R187 (2002)

    Article  MathSciNet  Google Scholar 

  12. Piotrowski, E.W., Sladkowski, J.: An invitation to quantum game theory. Int. J. Theor. Phys. 42(5), 1089–1099 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Guo, H., Zhang, J., Koehler, G.J.: A survey of quantum games. Decis. Support Syst. 46, 318–332 (2008)

    Article  Google Scholar 

  14. Khan, F.S., Phoenix, S.J.D.: Gaming the quantum. Quantum Inf. Comput. 13(3–4), 231–244 (2013)

    MathSciNet  Google Scholar 

  15. Khan, F.S., Phoenix, S.J.D.: Mini-maximizing two qubit quantum computations. Quantum Inf. Process. 12(12), 3807–3819 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Rezakhani, A.T.: Characterization of two-qubit perfect entanglers. Phys. Rev. A 70, 052313 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Vidal, G., Dawson, C.M.: Universal quantum circuit for two-qubit transformations with three controlled-NOT gates. Phys. Rev. A 69, R010301 (2004)

    ADS  Google Scholar 

  18. Makhlin, Y.: Nonlocal properties of two-qubit gates and mixed states, and the optimization of quantum computations. Quantum Inf. Proc. 1, 243–252 (2002)

    Article  MathSciNet  Google Scholar 

  19. Zhang, J., Vala, J., Whaley, K.B., Sastry, S.: Geometric theory of nonlocal two-qubit operations. Phys. Rev. A 67, 042313 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  20. Shende, V.V., Bullock, S.S., Markov, I.L.: Recognizing small-circuit structure in two-qubit operators. Phys. Rev. A 70, 012310 (2004)

    Article  ADS  Google Scholar 

  21. Vatan, F., Williams, C.: Optimal quantum circuits for general two-qubit gates. Phys. Rev. A 69, 032315 (2004)

    Article  ADS  Google Scholar 

  22. Balakrishnan, S., Sankaranarayanan, R.: Entangling power and local invariants of two-qubit gates. Phys. Rev. A 82, 034301 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Balakrishnan, S., Sankaranarayanan, R.: Operator-Schmidt decomposition and the geometrical edges of two-qubit gates. Quantum Inf. Process. 10(4), 449–461 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Situ, H., Huang, Z.: Relativistic quantum Bayesian game under decoherence. Int. J. Theor. Phys. 55, 2354–2363 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Deng, X., Deng, Y., Liu, Q., Wang, Z.: Quantum games of opinion formation based on the Marinatto–Weber quantum game scheme. Eur. Phys. Lett. 114, 50012 (2016)

    Article  ADS  Google Scholar 

  26. Deng, X., Deng, Y., Liu, Q., Chang, S., Wang, X.: A quantum extension to inspection game. Eur. Phys. J. B 89, 162 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  27. Coffey, M.W., Deiotte, R., Semi, T.: Comment on “Universal quantum circuit for two-qubit transformations with three controlled-NOT gates” and “Recognizing small-circuit structure in two-qubit operators”. Phys. Rev. A 77, 066301 (2008)

    Article  ADS  Google Scholar 

  28. Coffey, M.W., Deiotte, R.: Exact canonical decomposition of two-qubit operators in terms of CNOT. Quantum Inf. Process. 9, 681–691 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Coffey, M.W., Deiotte, R.: Relation of operator Schmidt decomposition and CNOT complexity. Quantum Inf. Process. 7, 117–124 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge the anonymous reviewers for their critical suggestions, in particular pointing the references [14, 15] which certainly add value to our main results.

Author information

Authors and Affiliations

  1. Department of Physics, School of Advanced Sciences, VIT University, Vellore, 632014, India

    V. Vijayakrishnan & S. Balakrishnan

Authors
  1. V. Vijayakrishnan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Balakrishnan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Balakrishnan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vijayakrishnan, V., Balakrishnan, S. Correspondence between quantization schemes for two-player nonzero-sum games and CNOT complexity. Quantum Inf Process 17, 102 (2018). https://doi.org/10.1007/s11128-018-1870-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-018-1870-5

Keywords

Search

Navigation