本論文的主要目的是將非線性H_∞控制理論引入微觀世界的系統，從而建立一種新形式的非線性H_∞量子位元強健控制技術，作為H_∞控制與量子系統兩個領域的橋樑
。本論文於量子控制領域中，率先開發非線性H_∞量子控制，並將其應用至磁場操控的電子自旋耦合系統，透過其強健的功能，使得在外在環境的干擾下，仍然保有精確的量子位元切換效果。
目前國際上針對電子自旋的量子控制方法，均局限於單一電子自旋或雙電子自旋耦合的情形，本論文將討論一般性N個自旋電子耦合的情形，並將其表示成標準型式的input-affine 非線性系統，方便與現有非線性控制理論的結合，並藉由針對多電子自旋耦合系統的特殊雙線性結構，將Hamilton-Jacobi PDE方程式化簡成等義的非狀態相依Riccati方程式(state-independent algebraic Riccati equation)或是更簡易的標準Lyapunov方程式，來建立最佳化H_∞量子控制器。本論文並引入複數化外擾來詮釋量子系統的環境外擾現象，使得在外界環境干擾下，量子系統的行為可用動力學的方式來定量描述。最後以雙電子、三電子及四電子自旋來驗證非線性H_∞量子位元控制的強健性，同時檢測在各種環境外擾的作用下，非線性H_∞強健控制是否仍能確保位元切換功能。

Nonlinear H_∞ Robust Control of Qubit

Student: Yu-Sheng Chen
Advisor: Ciann-Dong Yang
Department of Aeronautics and Astronautics
National Cheng Kung University

SUMMARY

The main purpose of this thesis is to introduce the nonlinear H_∞ control theory into the microscopic world system, so as to establish a new form of nonlinear H_∞ quantum-bit (qubit) robust control technology as a bridge between H_∞ control and quantum system. For the first time in the field of quantum control, this thesis develops the nonlinear H_∞ quantum control and applies it to the electron-spin coupling system under the magnetic field control. Through the robust function of H_∞ control, the accuracy of qubit transfer can be maintained in the presence of environmental disturbances. At present, the existing quantum control method for electron spin is confined to the case of single or double electrons. This thesis will discuss the general case involving N spin-coupling electrons and express it as a standard input-affine nonlinear system in order to combine with the existing nonlinear H_∞ control theory. Taking advantage of the bilinear structure of the multi-electron spin coupling system, we reduce the Hamilton-Jacobi PDE to an equivalent state-independent Riccati equation, from which the nonlinear H_∞ quantum control can be derived analytically. This thesis also introduces the complex-valued disturbances to the quantum system, so that the behavior of the controlled quantum system can be described dynamically under the action of external disturbances. At the end of the paper, the robustness of nonlinear H_∞ qubit control is verified by double electron spin to check whether the nonlinear H_∞ robust control can ensure the qubit transfer performance in the presence of various external disturbances.

Keywords: Quantum Control, Lyapunov Control, H_∞ Control, Nonlinear Control.

INTRODUCTION

The main purpose of this thesis is to introduce the nonlinear H_∞ control theory into the microscopic world system, so as to establish a new form of nonlinear H_∞ quantum-bit (qubit) robust control technology as a bridge between H_∞ control and quantum system.
At the end of the paper, the robustness of nonlinear H_∞ qubit control is verified by double electron spin to check whether the nonlinear H_∞ robust control can ensure the qubit transfer performance in the presence of various external disturbances.

MATERIALS AND METHODS

For the first time in the field of quantum control, this thesis develops the nonlinear H_∞ quantum control and applies it to the electron-spin coupling system under the magnetic field control. Through the robust function of H_∞ control, the accuracy of qubit transfer can be maintained in the presence of environmental disturbances. At present, the existing quantum control method for electron spin is confined to the case of single or double electrons. This thesis will discuss the general case involving N spin-coupling electrons and express it as a standard input-affine nonlinear system in order to combine with the existing nonlinear H_∞ control theory. Taking advantage of the bilinear structure of the multi-electron spin coupling system, we reduce the Hamilton-Jacobi PDE to an equivalent state-independent Riccati equation, from which the nonlinear H_∞ quantum control can be derived analytically. This thesis also introduces the complex-valued disturbances to the quantum system, so that the behavior of the controlled quantum system can be described dynamically under the action of external disturbances.

RESULT AND DISCUSSION

Use an equivalent state-independent Riccati equation, from which the nonlinear H_∞ quantum control can be derived analytically. The control qubit can be maintained the balance point as following results. Suppose the balance point is two-electron state -0.5i|├ 00⟩┤+0.5i|0├ 1⟩┤-0.5|├ 10⟩┤-0.5|├ 11⟩┤ under the action of external disturbances, simulation results are as follows.

Figure 1. Population of two-electron H_∞ robust control

Figure 2. The magnetic field control of two-electron H_∞ robust control

From Figure 1 and Figure 2, the input of the external disturbance causes the concussion of the same type of population, and the magnetic field control take population around the balance point, so you can see population up and down the frequency of the shock than the input of the larger. But also because of the external disturbance of the input, coupled with the moment can not be population control to 0, so you can see the magnetic field control is constantly changing.

CONCLUSION

From the simulation results, the new quantum H_∞ controller we have established is robust to the quantum bilinear systems we have established, and the extremely complex Hamilton-Jacobi PDE equation is reduced to simple The Lyapunov equation is convenient for future generations to solve.
Even if we discuss the general N spin-electron coupling cases and establish the quantum disturbance mathematical model of quantum systems, and express it as a standard type input-affine nonlinear system, and the existing nonlinear control Theory, but to actually simulate more than three electrons in the Matlab operation will spend more time.
1. When the external attenuation coefficient γ is fixed, the larger the weighting factor ρ, the smaller the size of the control field offset ‖u ̅(t)‖_2.
2. When the weighting factor ρ is fixed, the external disturbance coefficient γ is smaller and the control field offset ‖u ̅(t)‖_2 is smaller.
3. Even if the Lyapunov equation must have the solution of positive definite P, but this positive definite P does not necessarily satisfy the best quantum H_∞ controller, the critical value is determined by γ to γ_min.
4. We can see for any external disturbance has a certain degree of robustness.

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