Quasi non-Markovian Approach to the Study of Decoherence of a Controlled-Not Quantum Gate in a Chain of Few Nuclear Spins Quantum Computer

ABSTRACT

We develop in the weak coupling approximation a quasi-non-Markovian master equation and study the phenomenon of decoherence during the operation of a controlled-not (CNOT) quantum gate in a quantum computer model formed by a linear chain of three nuclear spins system with second neighbor Ising interaction between them. We compare with the behavior of the Markovian counterpart for temperature different from zero (thermalization) and at zero temperature for low and high dissipation rates. At low dissipation there is a very small difference between Markovian and quasi no-Markovian at any temperature which is unlikely to be measured, and at high dissipation there is a difference which is likely to be measured at any temperature.

We develop in the weak coupling approximation a quasi-non-Markovian master equation and study the phenomenon of decoherence during the operation of a controlled-not (CNOT) quantum gate in a quantum computer model formed by a linear chain of three nuclear spins system with second neighbor Ising interaction between them. We compare with the behavior of the Markovian counterpart for temperature different from zero (thermalization) and at zero temperature for low and high dissipation rates. At low dissipation there is a very small difference between Markovian and quasi no-Markovian at any temperature which is unlikely to be measured, and at high dissipation there is a difference which is likely to be measured at any temperature.

Cite this paper

P. López Vázquez and G. López Vázquez, "Quasi non-Markovian Approach to the Study of Decoherence of a Controlled-Not Quantum Gate in a Chain of Few Nuclear Spins Quantum Computer,"*Journal of Modern Physics*, Vol. 3 No. 9, 2012, pp. 902-917. doi: 10.4236/jmp.2012.39118.

P. López Vázquez and G. López Vázquez, "Quasi non-Markovian Approach to the Study of Decoherence of a Controlled-Not Quantum Gate in a Chain of Few Nuclear Spins Quantum Computer,"

References

[1] G. López, M. Murgua and M. Sosa, “Quantization of One-Dimensional Free Particle Motion with Dissipation,” Modern Physics Letters B, Vol. 15, No. 22, 2001, pp. 965-971. doi:10.1142/S0217984901002750

[2] G. López and P. López, “Velocity Quantization Approach of the One-Dimensional Dissipative Harmonic Oscillator,” International Journal of Theoretical Physics, Vol. 45, No. 4, 2006, pp. 734-742. doi:10.1007/s10773-006-9064-9

[3] H.-P. Breuer and F. Petruccione, “The Theory of Open Quantum Systems,” Oxford University Press, Oxford, 2006.

[4] G. Lindblad, “On the Generators of Quantum Dynamical Semigroups,” Communications in Mathematical Physics, Vol. 48, No. 2, 1976, pp. 119-130. doi:10.1007/BF01608499

[5] A. O. Caldeira and A. T. Legget, “Path Integral Approach to Quantum Brownian Motion,” Physica A, Vol. 121, No. 3, 1983, pp. 587-616. doi:10.1016/0378-4371(83)90013-4

[6] B. L. Hu, J. P. Paz and Y. Zhang, “Quantum Brownian Motion in a General Environment: Exact Master Equation with Nonlocal Dissipation and Colored Noised,” Physical Review D, Vol. 45, No. 8, 1992, pp. 2843-2861. doi:10.1103/PhysRevD.45.2843

[7] J. P. Paz and W. H. Zurek, “Environment-Induced Decoherence, Classicality and Consistency of Quantum Histories,” Physical Review D, Vol. 48, 1993, pp. 2728-2738. doi:10.1103/PhysRevD.48.2728

[8] A. Rivas, A. D. K. Plato, S. F. Huelga and M. B. Plenio, “Markovian Master Equations: A Critical Study,” New Journal of Physics, Vol. 12, 2010, Article ID: 113032. doi:10.1088/1367-2630/12/11/113032

[9] W. H. Zurek, “Decoherence, Einselection, and the Quantum Origins of the Classical,” Reviews of Modwen Physics, Vol. 75, 2003, pp. 715-775.

[10] W. H. Zurek, “Decoherence and the Transition from Quantum to Classical,” arXiv: quant-ph/0306072, 2003, pp. 1-24.

[11] H. D. Zeh, “There Is Not ‘First’ Quantization,” Physical Letters A, Vol. 309, No. 5, 2003, pp. 329-334. doi:10.1016/S0375-9601(03)00209-3

[12] M. Zwolak, H. T. Quan and W. H. Zurek, “Quantum Darwinism in a Mixed Environment,” Physical Review Letters, Vol. 103, 2009, Article ID: 110402. doi:10.1103/PhysRevLett.103.110402

[13] L. Mazzola, J. Piilo and S. Maniscalco, “Sudden Transition between Classical and Quantum Decoherence,” Physical Review Letters, Vol. 104, No. 20, 2010, Article ID: 200401. doi:10.1103/PhysRevLett.104.200401

[14] F. Intravaia, S. Maniscalco and A. Messina, “Density-Matrix Operatorial Solution of the Non-Markovian Master Equation for Quantum Brownian Motion,” Physical Review A, Vol. 67, No. 4, 2003, Article ID: 042108. doi:10.1103/PhysRevA.67.042108

[15] S. Maniscalco and F. Petruccione, “Non-Markovian Dynamics of a Qubit,” Physical Review A, Vol. 73, No. 1, 2006, Article ID: 012111. doi:10.1103/PhysRevA.73.012111

[16] H.-P. Breuer, “Non-Markovian Generalization of the Lindblad Theory of Open Quantum Systems,” Physical Review A, Vol. 75, No. 2, 2007, Article ID: 022103. doi:10.1103/PhysRevA.75.022103

[17] H.-P. Breuer, E.-M. Laine and J. Piilo, “Measure for the Degree of Non-Markovian Behavior of Quantum Processes in Open Systems,” Physical Review A, Vol. 103, No. 21, 2009, Article ID: 210401.

[18] A. Rivas, S. F. Huelga and M. B. Plenio, “Entanglement and Non-Markovian of Quantum Evolutions,” Physical Review Letters, Vol. 105, No. 5, 2010, Article ID: 050403. doi:10.1103/PhysRevLett.105.050403

[19] G. P. Berman, D. D. Doolen, D. I. Kamenev, G. V. López and V. I. Tsifrinovich, “Perturbation Theory and Numerical Modeling of Quantum Logic Operations with Large Number of Qubits,” Contemporary Mathematics, Vol. 305, 2002, pp. 13-41. doi:10.1090/conm/305/05213

[20] D. Solenov, D. Tolkunov and V. Privman, “Exchange Interaction, Entanglement, and Quantum Noise Due to Thermal Bosonic Field,” Physical Review B, Vol. 75, No. 3, 2007, Article ID: 035134. doi:10.1103/PhysRevB.75.035134

[21] A. A. Slutskin, K. N. Bratus, A. Bergvall and V. S. Shumeiko, “Non-Markovian Decoherence of a Two-Level System Weakly Coupled to a Bosonic Bath,” Europhysics Letters, Vol. 96, No. 4, 2011, Article ID: 40003. doi:10.1209/0295-5075/96/40003

[22] N. P. Oxtopy, A. Rivas, S. F. Huelga and R. Fazio, “Probing a Composite Spin-Boson Environment,” New Journal of Physics, Vol. 11, 2009, Article ID: 063028.

[23] G. V. López and L. Lara, “Numerical Simulation of a Controlled-Controlled-Not (CCN) Quantum Gate in a Chain of Three Interacting Nuclear Spins System,” Journal of Physics B: Atomic, Molecular and Optical Physics, Vol. 39, No. 18, 2006, pp. 3897-3904. doi:10.1088/0953-4075/39/18/019

[24] G. V. López, J. Quezada, G. P. Berman, D. D. Doolen, and V. I. Tsifrinovich, “Numerical Simulation of a Quantum Controlled-Not Gate Implemented on Four-Spin Molecules at Room Temperature,” Journal of Optics B: Quan- tum and Semiclassical Optics, Vol. 5, No. 2, 2003, pp. 184-189. doi:10.1088/1464-4266/5/2/311

[25] G. V. López, T. Gorin and L. Lara, “Simulation of Grover’s Quantum Search Algorithm in an Ising-Nuclear-Spin-Chain Quantum Computer with First-and-Second-Nearest-Neighbor Couplings,” Journal of Physics B: Atomic, Molecular and Optical Physics, Vol. 41, No. 5, 2008, Article ID: 055504.

[26] N. Y. Yao, et al., “Scalable Architecture for a Room Temperature Solid-State Quantum Information Processor,” arXiv: 1012.2864v1, 2002.

[27] F. W. Cummings, “Stimulated Emission of Radiation in a Single Mode,” Physical Review Letters, Vol. 140, No. 4A, 1965, p. A1051.

[28] G. V. López and P. López, “Study of Decoherence of Elementary Gates Implemented in a Chain of Few Nuclear Spins Quantum Computer Model,” Journal of Modern Physics, Vol. 3, No. 1, 2012, p. 85.

[29] S. Das and G. S. Agarwal, “Decoherence Effects in Interacting Qubits under the Influence of Various Environments,” Journal of Physics B: Atomic, Molecular and Optical Physics, Vol. 42, 2009, Article ID: 205502.

[1] G. López, M. Murgua and M. Sosa, “Quantization of One-Dimensional Free Particle Motion with Dissipation,” Modern Physics Letters B, Vol. 15, No. 22, 2001, pp. 965-971. doi:10.1142/S0217984901002750

[2] G. López and P. López, “Velocity Quantization Approach of the One-Dimensional Dissipative Harmonic Oscillator,” International Journal of Theoretical Physics, Vol. 45, No. 4, 2006, pp. 734-742. doi:10.1007/s10773-006-9064-9

[3] H.-P. Breuer and F. Petruccione, “The Theory of Open Quantum Systems,” Oxford University Press, Oxford, 2006.

[4] G. Lindblad, “On the Generators of Quantum Dynamical Semigroups,” Communications in Mathematical Physics, Vol. 48, No. 2, 1976, pp. 119-130. doi:10.1007/BF01608499

[5] A. O. Caldeira and A. T. Legget, “Path Integral Approach to Quantum Brownian Motion,” Physica A, Vol. 121, No. 3, 1983, pp. 587-616. doi:10.1016/0378-4371(83)90013-4

[6] B. L. Hu, J. P. Paz and Y. Zhang, “Quantum Brownian Motion in a General Environment: Exact Master Equation with Nonlocal Dissipation and Colored Noised,” Physical Review D, Vol. 45, No. 8, 1992, pp. 2843-2861. doi:10.1103/PhysRevD.45.2843

[7] J. P. Paz and W. H. Zurek, “Environment-Induced Decoherence, Classicality and Consistency of Quantum Histories,” Physical Review D, Vol. 48, 1993, pp. 2728-2738. doi:10.1103/PhysRevD.48.2728

[8] A. Rivas, A. D. K. Plato, S. F. Huelga and M. B. Plenio, “Markovian Master Equations: A Critical Study,” New Journal of Physics, Vol. 12, 2010, Article ID: 113032. doi:10.1088/1367-2630/12/11/113032

[9] W. H. Zurek, “Decoherence, Einselection, and the Quantum Origins of the Classical,” Reviews of Modwen Physics, Vol. 75, 2003, pp. 715-775.

[10] W. H. Zurek, “Decoherence and the Transition from Quantum to Classical,” arXiv: quant-ph/0306072, 2003, pp. 1-24.

[11] H. D. Zeh, “There Is Not ‘First’ Quantization,” Physical Letters A, Vol. 309, No. 5, 2003, pp. 329-334. doi:10.1016/S0375-9601(03)00209-3

[12] M. Zwolak, H. T. Quan and W. H. Zurek, “Quantum Darwinism in a Mixed Environment,” Physical Review Letters, Vol. 103, 2009, Article ID: 110402. doi:10.1103/PhysRevLett.103.110402

[13] L. Mazzola, J. Piilo and S. Maniscalco, “Sudden Transition between Classical and Quantum Decoherence,” Physical Review Letters, Vol. 104, No. 20, 2010, Article ID: 200401. doi:10.1103/PhysRevLett.104.200401

[14] F. Intravaia, S. Maniscalco and A. Messina, “Density-Matrix Operatorial Solution of the Non-Markovian Master Equation for Quantum Brownian Motion,” Physical Review A, Vol. 67, No. 4, 2003, Article ID: 042108. doi:10.1103/PhysRevA.67.042108

[15] S. Maniscalco and F. Petruccione, “Non-Markovian Dynamics of a Qubit,” Physical Review A, Vol. 73, No. 1, 2006, Article ID: 012111. doi:10.1103/PhysRevA.73.012111

[16] H.-P. Breuer, “Non-Markovian Generalization of the Lindblad Theory of Open Quantum Systems,” Physical Review A, Vol. 75, No. 2, 2007, Article ID: 022103. doi:10.1103/PhysRevA.75.022103

[17] H.-P. Breuer, E.-M. Laine and J. Piilo, “Measure for the Degree of Non-Markovian Behavior of Quantum Processes in Open Systems,” Physical Review A, Vol. 103, No. 21, 2009, Article ID: 210401.

[18] A. Rivas, S. F. Huelga and M. B. Plenio, “Entanglement and Non-Markovian of Quantum Evolutions,” Physical Review Letters, Vol. 105, No. 5, 2010, Article ID: 050403. doi:10.1103/PhysRevLett.105.050403

[19] G. P. Berman, D. D. Doolen, D. I. Kamenev, G. V. López and V. I. Tsifrinovich, “Perturbation Theory and Numerical Modeling of Quantum Logic Operations with Large Number of Qubits,” Contemporary Mathematics, Vol. 305, 2002, pp. 13-41. doi:10.1090/conm/305/05213

[20] D. Solenov, D. Tolkunov and V. Privman, “Exchange Interaction, Entanglement, and Quantum Noise Due to Thermal Bosonic Field,” Physical Review B, Vol. 75, No. 3, 2007, Article ID: 035134. doi:10.1103/PhysRevB.75.035134

[21] A. A. Slutskin, K. N. Bratus, A. Bergvall and V. S. Shumeiko, “Non-Markovian Decoherence of a Two-Level System Weakly Coupled to a Bosonic Bath,” Europhysics Letters, Vol. 96, No. 4, 2011, Article ID: 40003. doi:10.1209/0295-5075/96/40003

[22] N. P. Oxtopy, A. Rivas, S. F. Huelga and R. Fazio, “Probing a Composite Spin-Boson Environment,” New Journal of Physics, Vol. 11, 2009, Article ID: 063028.

[23] G. V. López and L. Lara, “Numerical Simulation of a Controlled-Controlled-Not (CCN) Quantum Gate in a Chain of Three Interacting Nuclear Spins System,” Journal of Physics B: Atomic, Molecular and Optical Physics, Vol. 39, No. 18, 2006, pp. 3897-3904. doi:10.1088/0953-4075/39/18/019

[24] G. V. López, J. Quezada, G. P. Berman, D. D. Doolen, and V. I. Tsifrinovich, “Numerical Simulation of a Quantum Controlled-Not Gate Implemented on Four-Spin Molecules at Room Temperature,” Journal of Optics B: Quan- tum and Semiclassical Optics, Vol. 5, No. 2, 2003, pp. 184-189. doi:10.1088/1464-4266/5/2/311

[25] G. V. López, T. Gorin and L. Lara, “Simulation of Grover’s Quantum Search Algorithm in an Ising-Nuclear-Spin-Chain Quantum Computer with First-and-Second-Nearest-Neighbor Couplings,” Journal of Physics B: Atomic, Molecular and Optical Physics, Vol. 41, No. 5, 2008, Article ID: 055504.

[26] N. Y. Yao, et al., “Scalable Architecture for a Room Temperature Solid-State Quantum Information Processor,” arXiv: 1012.2864v1, 2002.

[27] F. W. Cummings, “Stimulated Emission of Radiation in a Single Mode,” Physical Review Letters, Vol. 140, No. 4A, 1965, p. A1051.

[28] G. V. López and P. López, “Study of Decoherence of Elementary Gates Implemented in a Chain of Few Nuclear Spins Quantum Computer Model,” Journal of Modern Physics, Vol. 3, No. 1, 2012, p. 85.

[29] S. Das and G. S. Agarwal, “Decoherence Effects in Interacting Qubits under the Influence of Various Environments,” Journal of Physics B: Atomic, Molecular and Optical Physics, Vol. 42, 2009, Article ID: 205502.