論文題目(中文): 量子多位元糾纏控制
論文題目(英文): Entangled Control of Multiple Quantum Bits
研究生: 張秦瑋
指導教授: 楊憲東
本論文的目的是打破傳統量子位元單一磁場控制單一電子的限制，提出單一磁場控制多顆電子的構思，並提出以糾纏態為中間過渡態的想法，使得量子多位元控制得到真正意義上的實現。本論文將利用Lyapunov控制理論推導出保證系統漸進穩定的控制律，並重新改寫Liouville方程式，使得電腦可以輕易執行任意粒子數的量子多位元控制。本論文藉由Path programming與Schmidt分解的引入，發現了一個糾纏控制的新現象：當受控粒子數越多，控制場反而要越小，這是因為太大的控制增益，容易破壞粒子間之糾纏連結。基於此一新現象的發現，本論文分析比較了各種糾纏度的定量指標，以做為判斷量子糾纏控制好壞的依據。
此外，本論文設計了一系列的光子實驗證明所提糾纏控制構想的可行性，並於文末討論如何將量子多位元糾纏控制應用於量子通訊領域中，使得量子演算法不再只是數學模型上的建構，而是能用工程的手段加以實現。
關鍵詞：量子控制、糾纏控制、Lyapunov控制、Liouville方程式、量子通訊、糾纏熵、相對糾纏熵。

Student: Chin-Wei Chaung
Advisor: Ciann-Dong Yang
Department of Aeronautics and Astronautics,National Cheng Kung University
ABSTRACT
This thesis intends to conquer the limitation of traditional qubit control that a single magnetic field can only control a single electron, by putting forward the idea of controlling multiple electrons (multiple quantum bits) via entanglement among them by a single magnetic field. No matter what number of electrons is involved, we show that the spin motion of multiple electrons can be described by a redefined Liouville equation and controlled by a Lyapunov-based control theory. With the introduction of the methods of path programming and Schmidt decomposition, the thesis successfully controls multiple electrons into an entangled state which allows quantum information between the electrons to be interchanged so that multiple electrons can be controlled as a whole by a single magnetic field. The entangled state is found to be vulnerable by a high-gain control. Once entanglement is lost, control of multiple electrons by a single magnetic field becomes impossible, no matter how large the control gain is. This thesis proposes several quantitative indices of entanglement to evaluate the ability of a control strategy to preserve quantum entanglement.
In addition, the thesis designs a series of optical experiments to prove the feasibility of the proposed idea of entangled control. As a final contribution, the thesis applies entangled control of multiple qubits to the field of quantum communication to realize mathematical quantum algorithms by practical engineering methods.
Keywords: Quantum Control, Entangled Control, Lyapunov Control, Liouville Equation, Relative Entropy, Entanglement Entropy.

[1] Altafini, C. and Ticozzi, F., Modeling and Control of Quantum Systems:An Introduction, IEEE Transactios on Automatic Control, Vol. 57, No. 8, August, 2012.
[2] Amano, T., Yamauchi, S., Sugaya, T., Komori, K., Control of subband energy levels of quantum dots using InGaAs gradient composition strain-reducing layer, 10.1063/1.2218825, IEEE, 2009.
[3] Arieh, I., Hans, Z. Munthe-Kaas, Syvert P. Nørsett and Antonella Zanna, “Lie-Group Methods”, Cambridge University, Acta Numerica, 2005, pp. 1–148.
[4] Bhave, A., Montagne, E., Describing Quantum Circuits with Systolic Arrays, Application-specific Systems, Architectures and Processors (ASAP'06) 0-7695-2682-9/06, IEEE, 2006.
[5] BOTSINIS, P., SOON, X. N. and HANZO, L., Quantum Search Algorithms, Quantum Wireless,and a Low-Complexity Maximum Likelihood Iterative Quantum Multi-User Detector Design, IEEE Digital Object Identifier, 10 May, 2013.
[6] Cong S., Yang Fei, Convergent Control Strategy of State in Open Quantum Systems, 30th Chinese Control Conference, Yantai, China, 2 2-24July, 2011.
[7] Cong, S., Feng, X., Quantum System Schmidt Decomposition and Its Geometric Analysis, 27th Chinese Control Conference, Kunming,Yunnan, China, 16-18 July, 2008.
[8] Cong, S., Lou, Y., Yang, J., Kuang, S., A Convergent Control Strategy for Quantum Systems, National Key Basic Research Program under Grant No. 2011CBA00200.
[9] D’Alessandro, D. and Dahleh, M., Optimal Control of Two-Level Quantum Systems, IEEE Transactions on Automatic Control, Vol. 46, No. 6, June, 2001.
[10] David P. Divincenzo, 2000, “The Physical Implementation of Quantum Computation”, Fortschr. Phys. 48, pp. 771-783.
[11] Dong, D, Chen, Z., Research on Modeling and Simulation of Quantum Control Systems, 5th World Congress on Intelligent Control and Automation, Hangzhou, P.R. China 15-19 June, 2004.
[12] Dong, D. and Petersen, I. R., Sampled-Data Control of Two-level Quantum Systems Based on Sliding Mode Design, 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC),Orlando, FL, USA, 12-15 December, 2011.
[13] Dong, D., Petersen, I.R., Quantum control theory and applications: A survey, IET Control Theory Appl., Vol. 4, Iss. 12, pp. 2651– 2671 2651 doi: 10.1049/iet-cta.2009.0508, 2010.
[14] Dong, D., Petersen, I.R., Quantum control theory and applications: A survey, IET Control Theory Appl., Vol. 4, Iss. 12, pp. 2651– 2671 2651 doi: 10.1049/iet-cta.2009.0508, 2010.
[15] Gough, J., Belavkin, V.A. and Smolyanov, O.G., “Hamilton-Jacobi-Bellman Equations for Quantum Optimal Control”, DAYS on DIFFRACTION, 2006, pp.293-297.
[16] Hagouel, P. I., and Karafyllidis, I. G., Quantum Computers: Registers, Gates and Algorithms, 28th International Conference on Microelectronics, NIŠ, Serbia, 13-16 May, 2012.
[17] Hanson, R., Elzerman, J.M., Willems van Beveren, L.H., Electron Spin Qubits in Quantum Dots, 0-7803-8684-1/04/IEEE, 2004.
[18] Jonker, P., and Han, J., On Quantum & Classical Computing with Arrays of Superconducting Persistent Current Qubits, 0-7695-0740-9/00, IEEE, 2000.
[19] Kuang, S., Cong, S., and Lou, Y., Population Control of Quantum States Based on Invariant Subsets Under a Diagonal Lyapunov Function, Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, 16-18 December, 2009.
[20] Kurizki, G., and Vardi, A., Quantum-Zeno Control of Collisional Entanglement in a Bose-Josephson Junction, Third International Conference on Quantum, Nano and Micro Technologies, 2009.
[21] Kurniawan, I., Dirr, G. and Helmke, U., Controllability Aspects of Quantum Dynamics: A Unified Approach for Closed and Open Systems, IEEE Transactios on Automatic Control, Vol. 57, No. 8, August, 2012.
[22] Li, G. and Jia, Q., Tracking Control Laws for Some Quantum Systems Based on Nonlinear Dynamic Inversion Method, IEEE 978-1-4799-3197-2/14, 2014.
[23] Liu, J. and Cong, S., Trajectory Tracking of Quantum States Based on Lyapunov Method, 9th IEEE International Conference on Control and Automation (ICCA), Santiago, Chile, 19-21 December, 2011.
[24] Liu, J., Cong, S., Zhu, Y., Adaptive Trajectory Tracking of Quantum Systems, 12th International Conference on Control, Automation and Systems, in ICC, Jeju Island, Korea, 17-21 October, 2012.
[25] Lou, Y., Cong, S., Yang, J., Kuang, S., Path Programming Control Strategy of Quantum, IET Control Theory and Applications,3rd August, 2010.
[26] Malinovsky, V. S. and Sola, I. R, Quantum Control of Entanglement by Phase Manipulation of Time-Delayed Pulse Sequence, Phys. Rev. A 70, 042305, 8 October, 2004.
[27] Meng F. and Cong, S. and Kuang, S., Implicit Lyapunov Control of Multi-Control Hamiltonian Systems Based on State Distance, 10th World Congress on Intelligent Control and Automation,Beijing,China,6-8 July,2012.
[28] Meng F. and Cong, S., Implicit Lyapunov Control of Multi-Control Hamiltonians Systems Based On the State Error, World Academy of Science, Engineering and Technology International Journal of Mechanical, Aerospace, Industrial and Mechatronics Engineering Vol:7 No:7, 2013.
[29] Mimo X., Chen X., Research on Quantum-bit Error Correction Coding for Smart Grid Substation, IEEE International Conference on Green Computing and Communications and IEEE Internet of Things and IEEE Cyber, Physical and Social Computing, 2013.
[30] Nihtila, M., Open-Loop Control Design via Parametrization Applied in a Two-Level Quantum System Model, 4th International Symposium on Communications,Control and Signal Processing, ISCCSP 2010, Limassol, Cyprus, 3-5 March, 2010.
[31] Qi B., Guo L.,Comparisons between Quantum Open-Loop Control and Closed-Loop Control, 27th Chinese Control Conference, Kunming, Yunnan, China, 16–18 July, 2008.
[32] Ramos,R. C., Strauch, F. W., Johnson,P. R., Berkley, A. J., Xu, H., Gubrud, Capacitively Coupled Josephson Junctions:A Two-Qubit System, IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 13, NO. 2, JUNE, 2003.
[33] Robert N., Jaehoon J. and Philip H., Microwave Simulation of Grover's Quantum Search Algorithm, IEEE Antennas and Propagation Magazine, Vol. 48, No. 5, October, 2006.
[34] Silveira, H. B., Pereira, P. S., Silva, da and Rouchon, P., A Time-Periodic Lyapunov Approach for Motion Planning of Controllable Driftless Systems on SU(n), Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, Shanghai, 16-18 December, 2009.
[35] Ticozzi, F., Lucchese, R., Cappellaro, P. and Viola, L., Hamiltonian Control of Quantum Dynamical Semigroups: Stabilization and Convergence Speed, IEEE Transactios on Automatic Control, Vol. 57, No. 8, August, 2012.
[36] Wang, Q. F., Quantum Optimal Control of Nonlinear Dynamics Systems Described by Klein-Gordon-Schrodinger Equations, 2006 American Control Conference Minneapolis, Minnesota, USA, 14-16 June, 2006.
[37] Wang, Q. F., Quantum Optimal Control of Nuclei in the Presence of Perturbation in Electric Field, IET Control Theory and Applications, 1 December, 2008.
[38] Wang, X., and Schirmer, S. G., Analysis of Lyapunov Method for Control of Quantum States, IEEE Transactios on Automatic Control, Vol. 55, No. 10, October, 2010.
[39] Yang, C. D., “A Scientific Realization and Verification of Yin-Yang Theory: Complex-Valued Mechanics,” International Journal of Nonlinear Science, and Numerical Simulation, Vol. 11, 2010, pp. 135-156.
[40] Yang, C. D., “Introduction of Quantum System”, Department of Aeronautics and Astronautics, National Cheng Kung University, 2014, pp. 6.27-6.44.
[41] Yang, C. D., “Nonlinear System and control”, Department of Aeronautics and Astronautics, National Cheng Kung University, 2014, pp. 104-137.
[42] Yang, C. D., “Trajectory Interpretation of the Uncertainty Principle in 1D Systems Using Complex Bohmian Mechanics,” Physics Letters A, 372, 2008, pp. 6240-6253.
[43] Yang, C. D., and Kuo C. H., Quantum Modeling and Control of Josephson Junction by Quantum Hamilton Mechanics, 33rd Chinese Control Conference, 28-30 July, Nanjing, China, 2014.
[44] Yang, C. D., Complex Mechanics, Progress in Nonlinear Science, Vol. 1, Asian Academic Publisher, Hong Kong, 2010.
[45] Zhang, Q., Wang, W., Wang, L., Quantum System Control Based on Lyapunov Technology, 4th International Conference on Natural Computation, Zhengzhou, 2008.
[46] Zhao, S., Lin, H., Sun, J. and Xue, Z., Implicit Lyapunov Control of Closed Quantum Systems, Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, Shanghai, P.R. China, December 16-18, 2009.
[47] Zhu Y., Cong S., Liu J., “Dynamical Trajectory Tracking Control in Quantum Systems”, 31st Chinese Control Conference, Hefei, 25-27 July, 2012.
[48] 匡森，基于Lyapunov和最佳控制理論的量子系统控制方法的研究，中國科學技術大學博士學位論文，2007。
[49] 樓越升，量子系統控制場設計方法研究，中國科學技術大學博士學位論文，2010。
[50] 叢爽、匡森，糾纏態的探測與製備：雙自旋系統的糾纏態製備，載於量子系統控制理論與方法(pp. 273-278)，中國，中國科學技術大學出版社。