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系統識別號 U0026-0812200915404817
論文名稱(中文) 量子邏輯運算閘之最佳化控制研究
論文名稱(英文) Scheme for Implementing Quantum Logic Gate via Optimal Quantum Control Theorem
校院名稱 成功大學
系所名稱(中) 工程科學系碩博士班
系所名稱(英) Department of Engineering Science
學年度 98
學期 2
出版年 98
研究生(中文) 何儀輝
研究生(英文) Yi-Huei He
電子信箱 n9696146@mail.ncku.edu.tw
學號 n9696146
學位類別 碩士
語文別 中文
論文頁數 140頁
口試委員 指導教授-黃吉川
口試委員-謝金源
口試委員-林振森
口試委員-廖德祿
口試委員-陳俊良
中文關鍵字 量子控制  量子資訊  量子計算 
英文關鍵字 Quantum computation  Quantum Information  Quantum control 
學科別分類
中文摘要 本文將以雙原子分子C12O16內部振動和轉動運動行為來建1-qubit和2-qubits基本量子閘,從基於量子最佳化控制方案,控制分子系統在不同能階的躍遷,數值模擬找出一組量子閘作用的最佳輸入雷射場,其最佳化控制採用密度矩陣和劉維-馮·諾伊曼方程式描述多體量子系綜動力學演化,並結合投影算符將系統分為主要狀態和殘餘狀態兩個子空間,以提高演算法效率,模擬結果量子閘保真度和平均轉移機率均有96% 以上良率,提供實現量子計算中量子邏輯閘運算後量子訊息傳遞和儲存的路徑。
英文摘要 The motion behaviors of internal rovibration of a diatomic molecule are applied to build the elementary quantum gates encode one qubit and two qubits in this research. The molecular system we chose was carbon monoxide (C12O16). We used the optimal control theory to examined an optimization gate laser pulse numerically to control molecular system transition of different energy state. The controlled quantum systems are described with the probability-density-matrix based on Liouville–von Neumann equation. Projection operators were used to improve efficiency, and the states of the quantum system are decomposed into two sub-spaces, namely the ‘main state’ space and the ‘remaining state’ space. The numerical results show high gate fidelity and average transition probability over 96% to promise that it's reachable to perform transferable and storable paths of quantum information via distinguishably quantum logic gates.
論文目次 中文摘要 I
Abstract II
致謝 III
目錄 IV
表目錄 VII
圖目錄 VIII
符號說明 IX
第一章 緒論 1
1-1研究背景 1
1-2文獻回顧 5
1-3研究動機 9
1-4本文組織架構 10
第二章 量子統計力學與密度矩陣理論 12
2-1 量子統計力學 13
2-1-1 量子純態和量子混合態 14
2-1-2 密度矩陣 16
2-1-3 可觀測物理期望值 20
2-1-4 劉維爾-馮•諾伊曼方程式 22
2-2 約化劉維空間 24
2-2-1 劉維空間基底表示 24
2-2-2 可觀測物理期望值 27
2-2-3 劉維爾-馮•諾伊曼方程式 28
2-3 量子資訊基礎 32
2-3-1 量子位和量子邏輯閘 33
2-3-2 量子糾纏態 38
第三章 量子最佳化控制計劃 40
3-1 雙原子分子控制系統 41
3-1-1 轉振光譜能階解析解 42
3-1-2角動量耦合和Winger 3-j符號 45
3-1-3 狀態基底和相互作用哈密頓量 49
3-1-4 系統可控制性和可觀測性 52
3-2 量子最佳化控制 56
3-2-1 目標泛函確立 58
3-2-2 歐拉-拉格朗日方程式 60
3-2-3 單調收斂糾纏回授演算法 66
3-2-4 對稱分離算符實現么正演化算符 69
第四章 數值實現多維度量子封閉系統控制 75
4-1 投影算符法 76
4-2 控制場設計 83
4-3 數值模擬 91
第五章 結論與展望 107
5-1 結論 107
5-2 未來展望 108
參考文獻 110
附錄A-1 么正時間演化算符推導 122
附錄A-2 么正時間演化算符的數值方法 127
附錄B Morse能階 128
附錄C 數值演算李代數程式 134
附錄D 單位換算表 136
參考文獻 [1] C. H. Bennett and P. W. Shor, Quantum information theory, IEEE Transactions on information theory, Vol.44, No.6, pp.2724-2742, 1998.
[2] M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information , Cambridge University Press, Cambridge, U.K., 2000.
[3] W. Warren, H. Rabitz and M. Dahleh, Coherent control of quantum dynamics: The dream is alive, Science, Vol.259, No.5101, pp.1581-
1589, 1993.
[4] M. Shapiro and P. Brumer, Principle of the quantum control of molecular process, John Wiley & Sons Publ. Hoboken, New Jersey, 2003.
[5] T. Brixner and G. Gerber, Quantum control of gas-phase and liquid-phase femtochemistry, ChemPhysChem, Vol.4, No.5, pp.418-438, 2003.
[6] H. Rabitz, R. de Vivie-Riedle, M. Motzkus, and K. Kompa, Whither the future of controlling quantum phenomena?, Science, Vol.288, No.5467, pp.824-828, 2000.
[7] R. J. Levis, G. M. Menkir and H. Rabitz, Selective bond dissociation and rearrangement with optimally tailored, strong-field laser pulses, Science, Vol.292, No.5517 ,pp.709-713, 2001.
[8] R. P. Feynman, Quantum mechanical computers, Foundations of physics, Vol.16, No.6, pp.507-531, 1986.
[9] P. W. Shor, Algorithms for quantum computation: Discrete logarithms andfactoring, in proceedings of the symposium on the foundations of computer science, Los Alamitos, California, IEEE Computer society press, New York, pp.124-134, 1994.
[10] L. K. Grover, A fast quantum mechanics algorithm for database search, in proceedings of the twenty-eighth annual symposium on the theory of computing, Philadelphia, Pennsylvania, ACM Press, New York, pp.212-218, 1996.
[11] L. K. Grover, Quantum mechanics helps in searching for a needle in a haystack, Physical review letters, Vol.79, No.2, pp.325-328, 1997.
[12] T. Sleator and H. Weinfurter, Realizable universal quantum logic gates, Physical review letters, Vol.74, No.20, pp.4087-4090, 1995.
[13] M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raimond, and S. Haroche, Quantum rabi oscillation: A direct test of field quantization in a cavity, Physical review letters, Vol.76, No.11,pp.1800- 1803, 1996.
[14] J. I. Girac and P. Zoller, Quantum computations with cold trapped ions, Physical review letters, Vol.74, No.20, pp.4091-4094,1995.
[15] F. Schmidt-Kaler et al., Realization of the cirac–zoller controlled-
NOT quantum gate, Nature, Vol.422, No.6930, pp.408-411, 2003.
[16] N. A. Gershenfeld and I. L. Chuang, Spin-resonanse quantum computation, Science, Vol.275, No.5298, pp.350-356, 1997.
[17] J. A. Jones and M. Mosca, Implementation of a quantum algorithm on a nuclear magnetic resonance quantum computer, Journal of chemical physics, Vol.109, No.5, pp.1648-1653, 1998.
[18] R. Marx, A. F. Fahmy, J. M. Myers, W. Bermel and S. J. Glaser, Approaching five-bit NMR quantum computing, Physical review A, Vol.62, No.1, 012310, 2000.
[19] D. Loss and D.P. DiVincenzo, Quantum computation with quantum dots, Physical Review A, Vol.57, No.1, pp.120-126, 1998.
[20] L. M. K. Vandersypen, M. Steffen, G. Breyta, et al., Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance, Nature, Vol.414, No.6866, pp.883-887, 2001.
[21] A. O. Niskanen, K. Harrabi, F. Yoshihara, et al., Quantum coherent tunable coupling of superconducting qubits, Science, Vol.316, No.5825, pp.723-726, 2007.
[22] L. Viola, S. Lloyd and E. Knill, Universal control of decoupled quantum systems, Physical review letters, Vol.83, No.23, pp.4888-4891, 1999.
[23] R. S. Judson and H. Rabitz, Teaching lasers to control molecules, Physical review letters, Vol.68, No.10, pp.1500-1503, 1992.
[24] M. Q. Phan and H. Rabitz, Learning control of quantum-mechanical systems by laboratory identification of effective input-output maps, Chemical physics, Vol.217, No.2-3, pp.389-400, 1997.
[25] P. Gross, D. Neuhauser and H. Rabitz, Teaching lasers to control molecules in the presence of laboratory field uncertainty and measurement imprecision, Journal of chemical physics, Vol.98, No.6, pp.4557-4566, 1993.
[26] A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, Control of chemical reactions by feedback-
optimized phase-shaped femtosecond laser pulses, Science, Vol.282, No.5390, pp.919-922, 1998.
[27] J. W. Clark, C. K. Ong, T. J. Tarn and G. M. Huang, Quantum nondemolition filters, Mathematical systems theory, Vol.18, No.1, pp.33-55, 1985.
[28] H. M. Wiseman and G. J. Milburn, Quantum theory of optical feedback via homodyne detection, Physical review letters, Vol.70, No.5, pp.548-551, 1993.
[29] C. J. Bardeen, V. V. Yakovlev, K. R. Wilson, et al., Feedback quantum control of molecular electronic population transfer, Chemical physics letters, Vol.280, No.1-2, pp.151-158, 1997.
[30] A. C. Doherty and K. Jacobs, Feedback control of quantum systems using continuous state estimation, Physical Review A, Vol.60, No.4, pp.2700-2711, 1999.
[31] S. Lloyd, Coherent quantum feedback, Physical Review A, Vol.62, No.2, 022108, 2000.
[32] H. M. Wiseman, S. Mancini and J. Wang, Bayesian feedback versus Markovian feedback in a two-level atom, Physical Review A, Vol.66, No.1, 013807, 2002.
[33] J. M. Geremia, J. K. Stockton and H. Mabuchi, Real-time quantum feedback control of atomic spin-squeezing, Science, Vol.304, No.5668, pp.270-273, 2004.
[34] D. A. Steck, K. Jacobs, H. Mabuchi, T. Bhattacharya and S. Habib, Quantum feedback control of atomic motion in an optical cavity, Physical review letters, Vol.92, No.22, 223004, 2004.
[35] M. Yanagisawa, Quantum feedback control for deterministic entangled photon generation, Physical review letters, Vol.97, No.19, 190201, 2006.
[36] A. P. Peirce, M. A. Dahleh and H. Rabitz, Optimal control of quantum-mechanical systems: Existence, numerical approximation, and applications, Physical Review A, Vol.37, No.12, pp.4950 – 4964, 1988.
[37] J. M. Geremia and H. Rabitz, Optimal identification of Hamiltonian information by closed-loop laser control of quantum systems, Physical review letters, Vol. 89, No.26, 263902, 2002.
[38] A. C. Doherty, J. Doyle, H. Mabuchi, et al., Proceedings of the 39th IEEE conference on decision and control, Sydney, Australia, pp.949-954 , 2000.
[39] H. Y. Fan, K. Yang,D. M. Boye, et al., Self-assembly of ordered, robust, three-dimensional gold nanocrystal/silica arrays, Science, Vol.304, No.5670, pp.567-571, 2004.
[40] C. J. Bardeen, J. W. Che, K. R. Wilson, et al., Quantum control of NaI photodissociation reaction product states by ultrafast tailored light pulses, Journal of physical chemistry A, Vol.101, No.20, pp.3815-3822, 1997.
[41] C. J. Bardeen, J. W. Che, K. R. Wilson, et al., Quantum control of I2 in the gas phase and in condensed phase solid Kr matrix, Journal of chemical physics, Vol.106, No.20, pp.8486-8503, 1997.
[42] D. Meshulach and Y. Silberberg, Coherent quantum control of two-photon transitions by a femtosecond laser pulse. Nature, Vol.396, No.6708, pp.239-242 , 1998.
[43] S. M. Hurley and A. W. Castleman Jr., Laser chemistry: Keeping reactions under quantum control , Science, Vol.292, No.5517, pp. 648-649 (2001).
[44] C. M. Tesch, L. Kurtz, and R. de Vivie-Riedle, Applying optimal control theory for elements of quantum computation in molecular systems, Chemical physics letters, Vol.343, No.5-6, pp.633-641, 2001.
[45] S. Suzukia, K. Mishimab and K. Yamashita, Ab initio study of optimal control of ammonia molecular vibrational wavepackets: Towards molecular quantum computing, Chemical physics letters, Vol.410, No.4-6, pp.358-364, 2005.
[46] M. Tsubouchi and T. Momose, Rovibrational wave-packet manipulation using shaped midinfrared femtosecond pulses toward quantum computation: Optimization of pulse shape by a genetic algorithm, Physical Review A, Vol.77, No.5, 052326, 2008.
[47] K. Shioya, K. Mishima and K. Yamashita, Quantum computing using molecular vibrational and rotational modes, Molecular Physics Vol.105, No.9, pp.1283-1295, 2007.
[48] 詹政憲, 量子控制系統的可控制性研究, 中原大學碩士論文, 中壢, (1992).
[49] 陳弘暻, 量子系統的可控制性研究(二), 中原大學碩士論文, 中壢, (1993).
[50] 謝道明, 量子系統控制的理論研究, 中原大學碩士論文, 中壢, (2000).
[51] 周建良, 量子系統控制理論之研究, 成功大學碩士論文, 台南, (2003).
[52] C. J. Cheng, C. C. Hwang, T. L. Liao and G. L. Chou, Optimal control of quantum systems: a projection approach, Journal of physics A: mathematical and general, Vol.38, No.4, pp.929-942, 2005.
[53] U. Gaubatz, P. Rudecki, S. Schiemann and K. Bergmann, Population transfer between molecular vibrational levels by stimulated Raman scattering with partially overlapping laserfields. A new concept and experimental results, Journal of chemical physics, Vol.92, No.9, pp.5363-5376, 1990.
[54] G. W. Coulston and K. Bergmann, Population transfer by stimulated Raman scattering with delayed pulses: Analytical results for multilevel systems, Journal of chemical of physics, Vol.96, No.5, pp.3467-3475 , 1992.
[55] B. Glushko and B. Kryzhanovsky, Radiative and collisional damping effects on efficient population transfer in a three-level system driven by two delayed laser pulses, Physical review A, Vol.46, No.5, pp.2823-2830, 1992.
[56] M. Morillo, and R. I. Cukier, Control of proton-transfer reactions with external fields, Journal of chemical physics, Vol.98, No.6, pp.4548-4557, 1993.
[57] D. Y. Petrina, Mathematical foundations of quantum statistical mechanics, Kluwer Academic Pub., 1995.
[58] R. Shankar, Principles of quantum mechanics, Plenum Press, New York, 2nd ed., 1994.
[59] K. Blum, Density matrix theory and applications, physics of atoms and molecules, Springer, 2nd ed., 1996.
[60] J. von Neumann, The mathematical foundations of quantum mechanics, Princeton University Press, 1996.
[61] Y. Ohtsuki and Y. Fujimura, Bath-induced vibronic coherence transfer effects on femtosecond time-resolved resonant light scattering spectra from molecules, Journal of chemical physics, Vol.91, No.7, pp.3903-3915, 1989.
[62] J. Seke, A. V. Soldatov and N. N. Bogolubov Jr., Novel technique for quantum-mechanical eigenstate and eigenvalue calculatiions based on Seke's self-consistent projection-operator method, Modern physics letters. B, condensed matter physics, statistical physics, applied physics, Vol.11, No.6, pp.245-258, 1997.
[63] C. Uchiyama and F. Shibata, Unified projection operator formalism in nonequilibrium statistical mechanics, Physical review E, Vol.60, No.3, pp.2636, 1999.
[64] 叢爽, 量子力學系統控制導論, 科學出版社, 北京, (2006).
[65] S. G. Schirmer, Theory of control of quantum systems, UMI, Ann Arbor, MI, 2000.
[66] W. Zhu and H. Rabitz, A rapid monotonically convergent iteration algorithm for quantum optimal control over the expectation value of a positive definite operator, Journal of chemical physics, Vol.109, No.2, pp. 385-391, 1998.
[67] P. M. Morse, Diatomic molecules according to the wave mechanics. II. Vibrational levels. Physical review, Vol.34, No.1, pp.57-64, 1929.
[68] M. E. Goggin and P. W. Milonni, Driven Morse oscillator: Classical chaos, quantum theory, and photodissociation, Physical review A, Vol. 37, No.3, pp.796-806, 1988.
[69] H. Lefebvre-Brion and R. W. Field, The spectra and dynamics of diatomic molecules, ELSEVIER, 2004.
[70] J. Zúñiga, A. Bastida and A. Requena, An analytical perturbation treatment of the rotating Morse oscillator, Journal of physics B-atomic molecular and optical physics, Vol.41, No.10, 105102, 2008.
[71] J. O. Hirschfelder, C. F. Curtis, and R. B. Bird, Molecular theory of gases and liquids, Wiley, New York, 1954.
[72] S. G. Schirmer, H. Fu and A. I. Solomon, Complete controllability of quantum systems, Physical review A, Vol.63, No.6, 063410, 2001.
[73] S. G. Schirmer, H. Fu and A. I. Solomon, Complete controllability of finite-level quantum systems, Journal of physics A: mathematical and general, Vol.34, No.12, pp.1679-1690, 2001.
[74] A. G. Butkovskiy and Yu. I. Samoilenko, Control of quantum-
mechanical processes and systems, Kluwer Academic Pub., Dordrecht, Netherlands, 1990.
[75] A. E. Bryson, Jr and Yu Chi Ho, Applied optimal control: Optimization estimation and control, Hemisphere Pub., Washington, 1975.
[76] E. B. Lee and L. Markus, Foundations of optimal control theory, Wiley Pub., New York, 1967.
[77] A. P. Sage and C. C. White III, Optimum systems control, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1977.
[78] K. Sundermann and R. de Vivie-Riedle, Extensions to quantum optimal control algorithms and applications to special problems in state selective molecular dynamics, Journal of chemical physics, Vol.110, No.4, pp.1896-1904, 1999.
[79] G. Leitmann, The calculus of variations and optimal control, Plenum Press Pub., New York, 1981.
[80] S. Shi and H. Rabitz, Optimal control of bond selectivity in unimolecular reactions, Computer physics communications, Vol.63, No.1-3, pp.71-83, 1991.
[81] R. Kosloff, A.D. Hammerich and D. Tannor, Excitation without demolition: Radiative excitation of ground-surface vibration by impulsive stimulated Raman scattering with damage control, Physical review letters, Vol.69, No.15, pp.2172-2175, 1992.
[82] S. G. Schirmer, M. D. Girardeau, J. V. Leahy, Efficient algorithm for optimal control of mixed-state quantum systems, Physical review A, Vol.61, No.1, 012101, 2000.
[83] M. Suzuki, Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations, Physics letters A, Vol.146, No.6, pp.319-323, 1990.
[84] S. A. Rice and M. Zhao, Optical control of molecular dynamics, John Wiley & Sons Publ., New York, 2000.
[85] C. Uchiyama and F. Shibata, A systematic projection operator formalism in nonequilibrium statistical mechanics, Journal of the physical society of Japan, Vol.65, No.4, pp.887-890, 1996.
[86] H. P. Breuer and B. Kappler, The time-convolutionless projection operator technique in the quantum theory of dissipation and decoherence, Annals of physics, Vol.291, No.1, pp.36-70, 2001.
[87] J. Seke, A. V. Soldatov and N. N. Bogolubov Jr, The Seke self-consistent projection-operator approach for the calculation of quantum-mechanical eigenvalues and eigenstates, Physica A, Vol.246, No.1, pp.221-240, 1997.
[88] J. Seke, Self-consistent projection-operator method for describing the non-Markovian time-evolution of subsystems, Journal of physics A: mathematical and general, Vol.23, No.2, pp.L61, 1990.
[89] V. May and O. Kuhn, Charge and energy transfer dynamics in molecular systems, 2nd, Revised and Enlarged Edition, Wiley, 2004.
[90] J. Seke, A. V. Soldatov and N. N. Bogolubov Jr., Novel Technique for Quantum-Mechanical Eigenstate and Eigenvalue Calculatiions Based on Seke's Self-Consistent Projection-Operator Method, Mod. Phys. Lett. B 11(6), 245, (1997).
[91] O. Bayrak and I. Boztosun, Arbitrary l-state solutions of the rotating Morse potential by the asymptotic iteration method, Journal of physics A: mathematical and general, Vol.39, No.22, pp.6955–6963, 2006.
[92] H. Sekino and R. J. Bartlett, Molecular hyperpolarizabilities, J. Chem. Phys. 98, 3022 (1993).
[93] J. P. Palao and R. Kosloff, Optimal control theory for unitary transformations, Phys. Rev. A 68, 062308(2003).
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