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 |
Anti-Jaynes-Cummings, Jaynes-Cummings, Hadamard, Controlled-NOT
[1] | Nielsen, M. A. and Chuang, I. L. (2011) Quantum Computation and Quantum Information, Cambridge University Press. |
[2] | W. Scherer, W. (2019) Mathematics of Quantum Computing: An Introduction. Springer Nature. |
[3] | Deutsch, D. E. (1989) Quantum computational networks. Proc R. Soc. London A, 425 (1868): 73–90. |
[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. |
[6] | Barenco, A., Bennett, C. H., Cleve, R., DiVincenzo, D. P., Margolus, N., Shor, P., Sleator, T., Smolin, J. A., and Weinfurter, H. (1995) Elementary gates for quantum computation. Phys. Rev. A, 52 (5): 3457. |
[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. |
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
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
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
@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} }
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 -