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Theoretical Realization of a Two Qubit Quantum Controlled-NOT Logic Gate and a Single Qubit Quantum Hadamard Logic Gate in the Anti-Jaynes-Cummings Model

Received: 30 August 2021     Accepted: 22 September 2021     Published: 5 November 2021
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Abstract

Quantum gates are fundamental in Quantum computing for their role in manipulating elementary information carriers referred to as quantum bits. In this paper, a theoretical scheme for realizing a quantum Hadamard and a quantum controlled-NOT logic gates operations in the anti-Jaynes-Cummings interaction process is provided. Standard Hadamard operation for a specified initial atomic state is achieved by setting a specific sum frequency and photon number in the normalized anti-Jaynes-Cummings qubit state transition operation with the interaction component of the anti-Jaynes-Cummings Hamiltonian generating the state transitions. The quantum controlled-NOT logic gate is realized when a single atomic qubit defined in a two-dimensional Hilbert space is the control qubit and two non-degenerate and orthogonal polarized cavities defined in a two-dimensional Hilbert space make the target qubit. With precise choice of interaction time in the anti-Jaynes-Cummings qubit state transition operations defined in the anti-Jaynes-Cummings sub-space spanned by normalized but non-orthogonal basic qubit state vectors, ideal unit probabilities of success in the quantum controlled-NOT operations is determined.

Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 7, Issue 4)
DOI 10.11648/j.ijamtp.20210704.13
Page(s) 105-111
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2021. Published by Science Publishing Group

Keywords

Anti-Jaynes-Cummings, Jaynes-Cummings, Hadamard, Controlled-NOT

References
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[4] Raussendorf, R. and Briegel, H. J. (2001) A one-way quantum computer. Phys. Rev. Lett., 86 (22): 5188.
[5] Deutsch, D. and Jozsa, R. (1992) Rapid solution of problems by quantum computation. Proc. R. Soc. London A, 439 (1907): 553–558.
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[7] Shi, Y. (2001) Both Toffoli and controlled-NOT need little help to do universal quantum computation. arXiv preprint quant-ph/0205115.
[8] Boykin, P. O., Mor, T., Pulver, M., Roychowdhury, V. and Vatan, F. (2000) A new universal and fault-tolerant quantum basis. Inf. Process. Lett., 75 (3): 101–107.
[9] Omolo, J. A. (2021) Conserved excitation number and U (1) -symmetry operator for the anti-rotating (anti-Jaynes-Cummings) term of the Rabi Hamiltonian. arXiv preprint arXiv: 2103.06577 (Unpublished).
[10] Omolo, J. A. (2017) Polariton and anti-polariton qubits in the Rabi model. Preprint: Research Gate, DOI: 10.13140/ RG.2.2. 11833.67683 (Unpublished).
[11] Omolo, J. A. (2019) Photospins in the quantum Rabi model. Preprint: Research Gate, DOI: 10.13140/RG.2.2.27331.96807 (Unpublished).
[12] Barenco, A., Deutsch, D., Ekert, A., and Richard Jozsa, R. (1995) Conditional quantum dynamics and logic gates. Phys. Rev. Lett., 74 (20): 4083.
[13] Knill, E., Laflamme, R., Barnum, H., Dalvit, D., Dziarmaga, J., Gubernatis, J., Gurvits, L., Ortiz, G., Viola, L., and Zurek, W. H. (2002) Introduction to quantum information processing. arXiv preprint quant-ph/0207171.
[14] Braunstein, S. L., Mann, A., and Revzen, M. (1992) Maximal violation of Bell inequalities for mixed states. Phys. Rev. Lett., 68 (22): 3259.
[15] Domokos, P., Raimond, J.-M. Brune, M., and Haroche, S. (1995) Simple cavity-QED two-bit universal quantum logic gate: The principle and expected performances. Phys. Rev. A, 52 (5): 3554.
[16] Vitali, D., Giovannetti, V., and Tombesi. P. (2001) Quantum Gates and Networks with Cavity QED Systems. In Macroscopic Quantum Coherence and Quantum Computing, pages 235–244. Springer.
[17] Saif, F., Ul Islam, R. and Javed, M. (2007) Engineering quantum universal logic gates in electromagnetic-field modes. J. Russ. Laser Res., 28 (5): 529–534.
[18] Rossatto, D. Z., Villas-Boas, C. J., Sanz, M., and Solano, E. (2017) Spectral classification of coupling regimes in the quantum Rabi model. Phys. Rev. A 96, 013849.
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  • APA Style

    Christopher Mayero, Joseph Akeyo Omolo, Onyango Stephen Okeyo. (2021). Theoretical Realization of a Two Qubit Quantum Controlled-NOT Logic Gate and a Single Qubit Quantum Hadamard Logic Gate in the Anti-Jaynes-Cummings Model. International Journal of Applied Mathematics and Theoretical Physics, 7(4), 105-111. https://doi.org/10.11648/j.ijamtp.20210704.13

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    ACS Style

    Christopher Mayero; Joseph Akeyo Omolo; Onyango Stephen Okeyo. Theoretical Realization of a Two Qubit Quantum Controlled-NOT Logic Gate and a Single Qubit Quantum Hadamard Logic Gate in the Anti-Jaynes-Cummings Model. Int. J. Appl. Math. Theor. Phys. 2021, 7(4), 105-111. doi: 10.11648/j.ijamtp.20210704.13

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    AMA Style

    Christopher Mayero, Joseph Akeyo Omolo, Onyango Stephen Okeyo. Theoretical Realization of a Two Qubit Quantum Controlled-NOT Logic Gate and a Single Qubit Quantum Hadamard Logic Gate in the Anti-Jaynes-Cummings Model. Int J Appl Math Theor Phys. 2021;7(4):105-111. doi: 10.11648/j.ijamtp.20210704.13

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  • @article{10.11648/j.ijamtp.20210704.13,
      author = {Christopher Mayero and Joseph Akeyo Omolo and Onyango Stephen Okeyo},
      title = {Theoretical Realization of a Two Qubit Quantum Controlled-NOT Logic Gate and a Single Qubit Quantum Hadamard Logic Gate in the Anti-Jaynes-Cummings Model},
      journal = {International Journal of Applied Mathematics and Theoretical Physics},
      volume = {7},
      number = {4},
      pages = {105-111},
      doi = {10.11648/j.ijamtp.20210704.13},
      url = {https://doi.org/10.11648/j.ijamtp.20210704.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20210704.13},
      abstract = {Quantum gates are fundamental in Quantum computing for their role in manipulating elementary information carriers referred to as quantum bits. In this paper, a theoretical scheme for realizing a quantum Hadamard and a quantum controlled-NOT logic gates operations in the anti-Jaynes-Cummings interaction process is provided. Standard Hadamard operation for a specified initial atomic state is achieved by setting a specific sum frequency and photon number in the normalized anti-Jaynes-Cummings qubit state transition operation with the interaction component of the anti-Jaynes-Cummings Hamiltonian generating the state transitions. The quantum controlled-NOT logic gate is realized when a single atomic qubit defined in a two-dimensional Hilbert space is the control qubit and two non-degenerate and orthogonal polarized cavities defined in a two-dimensional Hilbert space make the target qubit. With precise choice of interaction time in the anti-Jaynes-Cummings qubit state transition operations defined in the anti-Jaynes-Cummings sub-space spanned by normalized but non-orthogonal basic qubit state vectors, ideal unit probabilities of success in the quantum controlled-NOT operations is determined.},
     year = {2021}
    }
    

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    AU  - Christopher Mayero
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    AB  - Quantum gates are fundamental in Quantum computing for their role in manipulating elementary information carriers referred to as quantum bits. In this paper, a theoretical scheme for realizing a quantum Hadamard and a quantum controlled-NOT logic gates operations in the anti-Jaynes-Cummings interaction process is provided. Standard Hadamard operation for a specified initial atomic state is achieved by setting a specific sum frequency and photon number in the normalized anti-Jaynes-Cummings qubit state transition operation with the interaction component of the anti-Jaynes-Cummings Hamiltonian generating the state transitions. The quantum controlled-NOT logic gate is realized when a single atomic qubit defined in a two-dimensional Hilbert space is the control qubit and two non-degenerate and orthogonal polarized cavities defined in a two-dimensional Hilbert space make the target qubit. With precise choice of interaction time in the anti-Jaynes-Cummings qubit state transition operations defined in the anti-Jaynes-Cummings sub-space spanned by normalized but non-orthogonal basic qubit state vectors, ideal unit probabilities of success in the quantum controlled-NOT operations is determined.
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Author Information
  • Department of Physics and Materials Science, Faculty of Science, Maseno University, Maseno, Kenya

  • Department of Physics and Materials Science, Faculty of Science, Maseno University, Maseno, Kenya

  • Department of Physics and Materials Science, Faculty of Science, Maseno University, Maseno, Kenya

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