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

Christopher Mayero; Joseph Akeyo Omolo; Onyango Stephen Okeyo

doi:doi:10.11648/j.ijamtp.20210704.13

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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

Christopher Mayero, Joseph Akeyo Omolo, Onyango Stephen Okeyo

Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 7, Issue 4)

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), 2024. Published by Science Publishing Group

Keywords

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

References

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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.
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[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.
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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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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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TY  - JOUR
T1  - 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
AU  - Christopher Mayero
AU  - Joseph Akeyo Omolo
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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.
VL  - 7
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Author Information

Christopher Mayero

Department of Physics and Materials Science, Faculty of Science, Maseno University, Maseno, Kenya
Joseph Akeyo Omolo

Department of Physics and Materials Science, Faculty of Science, Maseno University, Maseno, Kenya
Onyango Stephen Okeyo

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

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Plain Text BibTeX RIS

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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
T1  - 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
AU  - Christopher Mayero
AU  - Joseph Akeyo Omolo
AU  - Onyango Stephen Okeyo
Y1  - 2021/11/05
PY  - 2021
N1  - https://doi.org/10.11648/j.ijamtp.20210704.13
DO  - 10.11648/j.ijamtp.20210704.13
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 105
EP  - 111
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20210704.13
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.
VL  - 7
IS  - 4
ER  -

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