Study of Decoherence of Elementary Gates Implemented in a Chain of Few Nuclear Spins Quantum Computer Model

G. V. López, P. López

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DOI: 10.4236/jmp.2012.31013                

Abstract

We study the phenomenon of decoherence during the operation of one qubit transformation, controlled-not (CNOT) and controlled-controlled-not (C²NOT) quantum gates in a quantum computer model formed by a linear chain of three nuclear spins system. We make this study with different type of environments, and we determine the associated decoherence time as a function of the dissipative parameter. We found that the dissipation parameter to get a well defined quantum gates (without significant decoherence) must be within the range of . We also study the behavior of the purity parameter for these gates and different environments and found linear or quadratic decays of this parameter depending on the type of environments.

Keywords

Decoherence; Not Gate; Controlled-Not Gates; Controlled-Controlled-Not Gate; Nuclear Spin Chain; Quantum Computer

Conflicts of Interest

The authors declare no conflicts of interest.

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. 965-742. 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, 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]	A. J. Legget, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg and W. Zwerger, “Dynamics of the Dissipative Two-State System,” Reviews of Modern Physics, Vol. 59, No. 1, 1987, pp. 1-85. doi:10.1103/RevModPhys.59.1
[8]	W. G. Unruh and W. H. Zurek, “Reduction of a Wave Packet in Quantum Brownian Motion,” Physical Review D, Vol. 40, No. 4, 1989, pp. 1071-1094. doi:10.1103/PhysRevD.40.1071
[9]	A. Venugopalan, “Decoherence and Sch?dinger-Cat States in a Stern-Gerlach-Type Experiment,” Physical Review A, Vol. 56, No. 5, 1997, pp. 4307-4310. doi:10.1103/PhysRevA.56.4307
[10]	H. D. Zeh, “Toward Quantum Theory of Observation,” Foundations of Physics, Vol. 3, No. 1, 1973, pp. 109-116. doi:10.1007/BF00708603
[11]	J. P. Paz and W. H. Zurek, “Environment-Induced Decoherence, Classicality and Consistency of Quantum Histories,” Physical Review D, Vol. 48, No. 6, 1993, pp. 2728-2738. doi:10.1103/PhysRevD.48.2728
[12]	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, p. 113032. doi:10.1088/1367-2630/12/11/113032
[13]	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, p. 042108. doi:10.1103/PhysRevA.67.042108
[14]	S. Maniscalco and F. Petruccione, “Non-Markovian Dynamics of a Qubit,” Physical Review A, Vol. 73, No. 1, 2006, p. 012111. doi:10.1103/PhysRevA.73.012111
[15]	H.-P. Breuer, “Non-Markovian Generalization of the Lindblad Theory of Open Quantum Systems,” Physical Review A, Vol. 75, No. 2, 2007, p. 022103. doi:10.1103/PhysRevA.75.022103
[16]	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, 2009, p. 210401.
[17]	A. Rivas, S. F. Huelga and M. B. Plenio, “Entanglement and Non-Markovianity of Quantum Evolutions,” Physical Review Letters, Vol. 105, No. 5, 2010, p. 050403. doi:10.1103/PhysRevLett.105.050403
[18]	W. H. Zurek, “Decoherence, Einselection, and the Quantum Origins of the Classical,” Reviews of Modern Physics, Vol. 75, No. 3, 2003, pp. 715-775. doi:10.1103/RevModPhys.75.715
[19]	W. H. Zurek, “Decoherence and the Transition from Quantum to Classical,” 2003, pp. 1-24.
[20]	W. H. Zurek, “Decoherence and the Transition from Quantum to Classical,” Los Alamos Science, Vol. 27, 2002.
[21]	H. D. Zeh, “There Is Not ‘First’ Quantization,” Phys. Lett. A, Vol. 309, No. 5-6, 2003, pp. 329-334. doi:10.1016/S0375-9601(03)00209-3
[22]	M. Zwolak, H. T. Quan and W. H. Zurek, “Quantum Darwinism in a Mixed Environment,” Physical Review Letters, Vol. 103, No. 11, 2009, p. 110402. doi:10.1103/PhysRevLett.103.110402
[23]	L. Mazzola, J. Piilo and S. Maniscalco, “Sudden Transition between Classical and Quantum Decoherence,” Physical Review Letters, Vol. 104, No. 20, 2010, p. 200401. doi:10.1103/PhysRevLett.104.200401
[24]	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, p. 035134. doi:10.1103/PhysRevB.75.035134
[25]	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,” EPL, Vol. 96, No. 4, 2011, p. 40003. doi:10.1209/0295-5075/96/40003
[26]	N. P. Oxtopy, A. Rivas, S. F. Huelga and R. Fazio, “Probing a Composite Spin-Boson Environment,” New Journal of Physics, Vol. 11, 2009, p. 063028. doi:10.1088/1367-2630/11/6/063028
[27]	D. Cohen, J. von Deft, F. Marquardt and Y. Imry, “The Dephasing Rate Formula in the Many Body Context,” 2009.
[28]	Y. Hamdouni and F. Petruccione, “Time Evolution and Decoherence of a Spin- 1/2 Particle Coupled to a Spin Bath in Thermal Equilibrium,” Physical Review B, Vol. 76, No. 17, 2007, p. 174306. doi:10.1103/PhysRevB.76.174306
[29]	Z. Gedik, “Spin Bath Decoherence of Quantum Entanglement,” Solid State Communications, Vol. 138, No. 2, 2006, pp. 82-85. doi:10.1016/j.ssc.2006.02.004
[30]	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, No. 20, 2009, p. 205502. doi:10.1088/0953-4075/42/20/205502
[31]	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.
[32]	A. Shabani and D. A. Lindar, “Completely Positive Post-Markovian Master Equation via a Measurement Approach,” Physical Review A, Vol. 71, No. 2, 2005, p. 020101R. doi:10.1103/PhysRevA.71.020101
[33]	I. de Vega, D. Alonso and P. Gaspard, “Two-Level System Immersed in a Photonic Band-Gap Material: A Non-Markovian Stochastic Schr?dinger-Equation Approach,” Physical Review A, Vol. 71, No. 2, 2005, p. 023812. doi:10.1103/PhysRevA.71.023812
[34]	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
[35]	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: Quantum and Semiclassical Optics, Vol. 5, No. 2, 2003, pp. 184-189. doi:10.1088/1464-4266/5/2/311
[36]	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, p. 055504. doi:10.1088/0953-4075/41/5/055504
[37]	N. Y. Yao, et al., “Scalable Architecture for a Room Temperature Solid-State Quantum Information Processor,” 2002.
[38]	S. Lloyd, “A potential Realizable Quantum Computer,” Science, Vol. 261, No. 5128, 1993, pp. 1569-1571. doi:10.1126/science.261.5128.1569
[39]	F. H. L. Koppens, et al., “Driven Coherent Oscillations of a Single Electron Spin in a Quantum Dot,” Nature, Vol. 442, 2006, pp. 766-771. doi:10.1038/nature05065