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A Matrix Representation of Quantum Circuits over Non-Adjacent Qudits

  • Giuseppe SergioliEmail author
Article

Abstract

Within the general context of the architecture in quantum computer design, this paper aims is to provide a general strategy to obtain a block-matrix representation of quantum gates applied to qubits placed in arbitrary positions over an arbitrary dimensional input state. The model is also extended to the framework of quantum computation with qudits. An application in the context of the quantum computational logic is provided.

Keywords

Quantum circuit Block-matrix representation Quantum computational logic 

Notes

Acknowledgements

This work has been partially supported by Regione Autonoma della Sardegna in the framework of the project “Time-logical evolution of correlated microscopic systems” (CRP 55, L.R. 7/2007, 2015) and by Fondazione Banco di Sardegna in the framework of the project “Strategies and Technologies for Scientific Education and Dissemination”, cup: F71I17000330002.

References

  1. 1.
    Bernstein, D.: Matrix mathematics. Princeton University Press, Princeton (2005)Google Scholar
  2. 2.
    Broadbent, A., Kashefi, E.: Parallelizing quantum circuits. Theoret. Comput. Sci. 410(26, 6), 2489–2510 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bishop, L.S., Tornberg, L., Price, D., Ginossar, E., Nunnenkamp, A., Houck, A.A., Gambetta, J.M., Koch, J., Johansson, G., Girvin, S.M., Schoelkopf, R.J.: Proposal for generating and detecting multi-qubit GHZ states in circuit QED. New J. Phys. 11, 073040 (2009)ADSCrossRefGoogle Scholar
  4. 4.
    Castagnoli, G.: Completing the physical representation of quantum algorithms provides a quantitative explanation of their computational speedup. Found. Phys. 48(3), 333–354 (to appear)Google Scholar
  5. 5.
    Cheung, D., Maslov, D., Severini, S.: Translation techniques between quantum circuit architectures (2007)Google Scholar
  6. 6.
    Cohen, I., Weidt, S., Hensinger, W.K., Retzker, A.: Multi-qubit gate with trapped ions for microwave and laser-based implementation. New J. Phys. 17, 043008 (2015)ADSCrossRefGoogle Scholar
  7. 7.
    Dalla Chiara, M.L., Giuntini, R., Greechie, R.: Reasoning in quantum theory: sharp and unsharp quantum logic. Trends in Logic. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  8. 8.
    Dalla Chiara, M.L., Giuntini, R., Leporini, R., Sergioli, G.: Quantum computation and logic; how quantum computers have inspired logical investigations trends in logic n.48. Springer, Berlin (2018)CrossRefzbMATHGoogle Scholar
  9. 9.
    Dalla Chiara, M.L., Giuntini, R., Sergioli, G., Leporini, R.: A many-valued approach to quantum computational logic. Fuzzy Sets ans Systems 335, 94–111 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Freytes, H., Giuntini, R., Leporini, R., Sergioli, G.: Entanglement and quantum logical gates. Part I Int. J. Theor. Phys. 54(12), 3880–3888 (2015)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Fowler, A.G., Hill, C.D., Hollenberg, L.C.L.: Quantum error correction on linear neares neighbor qubit arrays. Phys. Rev. A 69, 042314.1–042314.4 (2004)ADSCrossRefGoogle Scholar
  12. 12.
    Fuchs, J.: Affine lie algebras and quantum groups. Cambridge University Press, Cambridge (1992)Google Scholar
  13. 13.
    Garcia-Escartin, J.C., Chamorro-Posada, P.: A SWAP gate for qudits. Quantum Inf. Process 12, 3625–3631 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gerdt, V.P., Prokopenya, A.N.: The circuit model of quantum computation and its simulation with mathematica, Mathematical Modelling and Computer Science, pp 43–55. LNCS-Springer, Berlin (2011)Google Scholar
  15. 15.
    Gerdt, V.P., Kragler, R., Prokopenya, A.N.: A Mathematica program for constructing quantum circuits and computing their unitary matrices. Phys. Part. Nucl. Lett. (Springer) 6, 526 (2009)CrossRefGoogle Scholar
  16. 16.
    Gerdt, V.P., Kragler, R., Prokopenya, A.N.: A mathematica package for simulation of quantum computation. In: Gerdt, V.P., Mayr, E.W., Vorozhtsov, E.V. (eds.) Computer Algebra in Scientific Computing. CASC 2009. Lecture Notes in Computer Science, vol. 5743. Springer, Berlin (2009)Google Scholar
  17. 17.
    Giuntini, R., Ledda, A., Sergioli, G., Paoli, F.: Some generalizations of fuzzy structures in quantum computational logic. Int. J. Gen. Syst. 40(1), 61–83 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Häffner, H., Hänsel, W., Roos, C.F., Benhelm, J., Chek al kar, D., Chwalla, M., Körber, T., Rapol, U.D., Riebe, M., Schmidt, P.O., Becher, C., Gühne, O., Dür, W., Blatt, R.: Scalable multipartite entanglement of trapped ions. Nature 438, 643–646 (2005)ADSCrossRefGoogle Scholar
  19. 19.
    Hirvensalo, M.: Quantum computing, natural computing series. Springer, Berlin (2001)zbMATHGoogle Scholar
  20. 20.
    Jozsa, R., Miyake, A.: Matchgates and classical simulation of quantum circuits. Proc. R. Soc. A 464, 3089–3106 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kane, B.: A solicon-based nuclear spin quantum computer. Nature 393, 133–137 (1998)ADSCrossRefGoogle Scholar
  22. 22.
    Kumar, P.: Efficient quantum computing between remote qubits in linear nearest neighbor architectures. Quantum Inf. Process 12-4, 1737–1757 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Laforest, M., Simon, D., Boileau, J.C., Baugh, J., Ditty, M., Laflamme, R.: Using error correction to determine the noise model. Phys. Rev. A 75, 133–137 (2007)CrossRefGoogle Scholar
  24. 24.
    Linke, N.M., Maslov, D., Roetteler, M., Debnath, S., Figgatt, C., Landsman, K. A., Wright, K., Monroe, C.: Experimental comparison of two quantum computing architectures. Proc. Natl. Acad. Sci. USA 114(13), 3305–3310 (2017)CrossRefGoogle Scholar
  25. 25.
    Möttönen, M., Vartiainen, J.J., Bergholm, V., Salomaa, M.M.: Quantum circuits for general multi-qubit gates. Phys. Rev. Lett. 93, 130502 (2004)CrossRefGoogle Scholar
  26. 26.
    Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  27. 27.
    Sergioli, G.: Towards a multi target quantum computational logic. Foundations of Sciences (to appear  https://doi.org/10.1007/s10699-018-9569-8)
  28. 28.
    Sergioli, G., Ledda, A.: A note on many valued quantum computational logics. Soft. Comput. 21, 1391–1400 (2017)CrossRefzbMATHGoogle Scholar
  29. 29.
    Sergioli, G., Leporini, R.: Quantum approach to epistemic semantics. Soft. Comput. 21-6, 1381–1390 (2017)CrossRefzbMATHGoogle Scholar
  30. 30.
    Takahashi, Y., Kunihiro, N., Ohta, K.: The quantum Fourier transform on a linear nearest neighbor architecture. Quantum Inf. Comput. 7, 383–391 (2007)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Thew, R.T., Nemoto, K., White, A.G., Munro, W.J.: Qudit quantum-state tomography. Phys. Rev. A 66, 012303 (2002)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Wilmott, C.M.: On swapping the states of two qudits. Int. J. Quant. Inf., vol. 9–1511 (2011)Google Scholar
  33. 33.
    Zhang, J., Liu, W., Deng, Z., Lu, Z., Lu Long, G.: Modularization of the multi-qubit controlled phase gate and its NMR implementation, arXiv:quant-ph/0406209v2 (2004)
  34. 34.
    Zhang, Z., Liu, Y., Wang, D.: Perfect teleportation of arbitrary n-qudit states using different quantum channels. Phys. Lett. A 372(1), 28–32 (2007)ADSCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of CagliariCagliariItaly

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