During recent years much work has been done to develop the quantum computers. In a quantum computer, the data is loaded as a string of quantum bits (qubits) [1]. Quantum gates perform very simple operations on these qubits such as flipping their values. By combining many quantum gates, complex operations can be realized and these operations can be used to manipulate the qubits. The preparation of quantum basic gates is one of the most important research topics in quantum control field [2]. The main objective is to prepare stable and highfidelity quantum gates within a possible short time and prevent them from decoherence as long as possible [3]. A quantum control process can be divided into coherent and decoherent parts, corresponding to the unitary and nonunitary operations, respectively [4], [5]. Up to now, many different quantum control methods have been developed to generate higher fidelity quantum gates in a short time. One of the common methods is the quantum optimal control method, which has been extensively studied [6][11]. Dynamical decoupling method is also an effective control way for the quantum gate preparation. In 2013, Piltz et al. protected conditional quantum gates by robust dynamical decoupling [12]. In 2011, Grace et al. combined dynamicaldecoupling pulses with the optimal control method for improving preparation of quantum gates [13]. However, in the methods mentioned above the control laws are not analytic and the designing procedure is a timeconsuming task. The design of control laws based on the quantum Lyapunov method greatly simplifies the mathematic calculation and its analytical type of control laws make the control system be easily adjusted [14], [15].
The Hadamard gate is one of the most basic and important gates in quantum computers [16]. Any unitary operation can be approximated with arbitrary accuracy by means of special gates set in which the Hadamard gate must be included. Many quantum algorithms use the Hadamard transformation as the first step to initialize the state with random information. In quantum information processing, the Hadamard transformation acts as a onequbit operator that maps the qubit basis states to different superposition states [17].
In our previous work [18] we prepared a Not gate for one qubit open quantum system. In this paper, we will design a Lyapunov control method to prepare the Hadamard gate using unitary timeevolution operator whose dynamics are transferred to the Bloch vector space. We construct a matrix logarithm function as the Lyapunov function. The design of control laws is based on the Lyapunov stability theorem. The purpose of the control is to drive the unitary evolution operator from any initial quantum gate as close as possible to the desired quantum gate in the shortest possible time. Two performance indices of the system under environment uncertainties are analyzed by means of the simulation experiments.
The rest of this paper is arranged as follows: in Section Ⅱ, the descriptions of the control system and the model of the system are studied. In Section Ⅲ, the Lyapunov function and the design of control laws are investigated. In Section Ⅳ, the Hadamard gate based on designed control laws is prepared in numerical experiments, the performances of control laws are analysed, and the comparisons with other control methods are done. Finally, the conclusion is given in Section Ⅴ.
Ⅱ. DESCRIPTIONS OF THE CONTROL SYSTEM AND THE MODEL OF THE SYSTEMFor a twolevel Markovian open quantum system, the dynamics of state
$ \dot{\rho_t }=i\left[H(t), \rho_t\right]+L(\rho_t) $  (1) 
where
$ H(t)=\ H_0+H_c $  (2) 
where
$ H_c=\frac{1}{2}\sum\limits_{k=x, y, z}{f_k(t)}{\sigma }_k $  (3) 
where
$ \begin{align} {\sigma }_x=\ \left[\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right], \ \ {\sigma }_y=\left[\begin{array}{cc} 0 &i \\ i & 0 \end{array} \right], \ \ {\sigma }_z=\left[ \begin{array}{cc} 1 & 0 \\ 0 &1 \end{array} \right]. \end{align} $  (4) 
In (1),
$ L\left(\rho_t\right)= \sum\limits_{\alpha, \beta }{\gamma } _{\alpha, \beta }\left[F_{\alpha }\rho_tF^{\dagger }_{\beta }\frac{1}{2}\left(F^{\dagger }_{\beta }\ F_{\alpha }\rho_t+\rho_tF^{\dagger }_{\beta }\ F_{\alpha }\right)\right] $  (5) 
where
$ \begin{align} \Gamma= \left[\! \begin{array}{ccc} {\gamma }_{xx} & {\gamma }_{xy} & {\gamma }_{xz} \\ {\gamma }_{yx} & {\gamma }_{yy} & {\gamma }_{yz} \\ {\gamma }_{zx} & {\gamma }_{zy} & {\gamma }_{zz} \end{array} \!\right]. \end{align} $  (6) 
In our work, the studied model of the Markovian open quantum system is amplitude damping (AD). The related GKS matrix for the AD system is [22], [23]
$ \begin{align} {\Gamma }_{AD}=\gamma \left[\begin{array}{ccc} 1 & i & 0 \\ i & 1 & 0 \\ 0 & 0 & 0 \end{array} \right]. \end{align} $  (7) 
Moreover, the dissipation part of the AD system is [23]
$ L_{AD}\left(\rho_t\right)= \frac{\gamma }{2}\left[{\sigma }_{i}\rho_t{\sigma }_{i+}\frac{1}{2}({\sigma }_{i+}{\sigma }_{i}\rho_t+\rho_t{\sigma }_{i+}{\sigma }_{i})\right] $  (8) 
where
The preparation of quantum gates is more comprehensible if they can be considered as a kind of operators. Under this consideration, the dynamics of the operators must be obtained. Since the density matrix dynamics of (1) is a bilinear equation with dissipation part, it is not easy to use to manipulate the gates. Fortunately for a twolevel quantum system, the state of the quantum system can also be described by the state vector.
As
$ \rho_t =\frac{1}{2}(I+r_{x_t}{\sigma }_x+r_{y_t}{\sigma }_y+r_{z_t}{\sigma }_z) $  (9) 
in this way
We define
$ {\rho }_f=U(t)\cdotp {\rho }_0\cdotp {U}^{\dagger }(t). $  (10) 
According to (1), (9) and (10), we can obtain the following dynamics equation
$ \dot{U}(t)=(A(t)+B)U(t) $  (11) 
in which
$ \begin{align}A(t)&=\left[\begin{array}{ccc} 0 &f_z(t) & f_y(t) \\ f_z(t) & 0 & {f}_x(t) \\ {f}_y(t) & f_x(t) & 0 \end{array} \right]\nonumber\\[1mm] &=f_x(t)A_x+f_y(t)A_y+f_z(t)A_z \end{align} $  (12) 
where
$ \begin{align} B=&\ \frac{\Gamma +{\Gamma}^T}{2}{\rm tr}\left(\Gamma\right)I\nonumber\\[1mm]=&\ \frac{1}{2} \left[\! \begin{array}{ccc} 2({\gamma }_{yy}+{\gamma }_{zz}) \!\!&\!\! {\gamma }_{xy}+{\gamma }_{yx} \!\!&\!\! {\gamma }_{xz}+{\gamma }_{zx} \\ {\gamma }_{yx}+{\gamma }_{xy} \!\!&\!\!2({\gamma }_{xx}+{\gamma }_{zz}) \!\!&\!\! {\gamma }_{yz}+{\gamma }_{zy} \\ {\gamma }_{zx}+{\gamma }_{xz} \!\!&\!\! {\gamma }_{zy}+{\gamma }_{yz} \!\!&\!\!2({\gamma }_{xx}+{\gamma }_{yy}) \end{array} \!\right]. \end{align} $  (13) 
Based on (6) and (7), for the AD system we set
$ \begin{align} B=\gamma \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 &1 & 0 \\ 0 & 0 &2 \end{array} \right]. \end{align} $  (14) 
From (9) and (10), the timeevolution of vector
$ \ r_t=U(t)\ \cdotp r_0. $  (15) 
Accordingly, based on (11) we can obtain
$ \dot{r}_t=\left(A(t)+B\right)r_t. $  (16) 
Now for preparing the quantum gate, the control task becomes to design the control fields in
By substituting the Pauli matrices (4) into the density matrix given by (9), the relationship between
$ \begin{align} {\rho_t }=\left[\begin{array}{ll} \dfrac{1}{2}(1+r_{z_t}) & \dfrac{1}{2}(r_{x_t}ir_{y_t}) \\[4mm] \dfrac{1}{2}(r_{x_t}+ir_{y_t}) & \dfrac{1}{2}(1r_{z_t}) \end{array} \right]. \end{align} $  (17) 
Consider the matrix
$ \begin{align} {\rho }_f=&\ { G}\cdotp {\rho }_0\cdotp {{ G}}^{\dagger } =\left[ \begin{array}{cc} u_1 & u_2 \\ u_3 & u_4 \end{array} \right]\nonumber\\[1mm] &\times\left[ \begin{array}{ll} \dfrac{1}{2}(1+r_{z{_{0}}}) & \dfrac{1}{2}(r_{x{_{0}}}ir_{y{_{0}}}) \\[2mm] \dfrac{1}{2}(r_{x{_{0}}}+ir_{y{_{0}}}) & \dfrac{1}{2}(1r_{z{_{0}}}) \end{array} \right]\left[\begin{array}{cc} {u_1}^* & {u_3}^* \\ {u_2}^* & {u_4}^* \end{array} \right]\nonumber\\[1mm] =&\ \frac{1}{2}I+\frac{1}{2}r_{x{_{0}}}\left[\begin{array}{cc} u^*_1u_2+u_1u^*_2 & u_2u^*_3+u_1u^*_4 \\ u^*_1u_4+u^*_2u_3 & u^*_3u_4+u_3u^*_4 \end{array} \right]\nonumber\\[1mm]& +\frac{i}{2}r_{y{_{0}}}\left[\begin{array}{cc} u^*_1u_2u_1u^*_2 & u_2u^*_3u_1u^*_4 \\ u^*_1u_4u^*_2u_3 & u^*_3u_4u_3u^*_4 \end{array} \right]\nonumber\\[1mm]& +\frac{1}{2}r_{z{_{0}}}\left[ \begin{array}{cc} {u_1u}^*_1u_2u^*_2 & u_1u^*_3u_2u^*_4 \\ u^*_1u_3u^*_2u_4 & u^*_3u_3u_4u^*_4 \end{array} \right]\end{align} $  (18) 
where
Let the final state vector be
$ \begin{align} r_{x{_f}}=&\ \frac{1}{2}\left(u^*_1u_4+u^*_2u_3+u_2u^*_3+u_1u^*_4\right)r_{x{_0}}\nonumber\\& +\frac{i}{2}\left(u_2u^*_3u_1u^*_4+u^*_1u_4u^*_2u_3\right) r_{y{_0}}\nonumber\\& + \frac{1}{2}\left(u^*_1u_3+{u_1u}^*_3u_2u^*_4+{u^*_2u}_4\right)r_{z{_0}} \end{align} $  (19) 
$ \begin{align} r_{y{_f}}=&\ \frac{i}{2}\left(u^*_1u_4u_1u^*_4+u_2u^*_3u^*_2u_3\right)r_{x{_0}}\nonumber\\& +\frac{1}{2}\left(u_1u^*_4+u^*_1u_4u^*_2u_3u_2u^*_3\right) r_{y{_0}} \nonumber\\& +\frac{i}{2}\left({u_1u}^*_3\ u^*_1u_3u_2u^*_4+{u^*_2u}_4\right)r_{z{_0}} \end{align} $  (20) 
$ \begin{align} r_{z{_f}}=&\ r_{x{_0}}\left(u^*_1u_2+u_1u^*_2\right)+ir_{y{_0}} \left(u^*_1u_2u_1u^*_2\right)\nonumber\\& +r_{z{_0}}\left({u_1u}^*_1u_2u^*_2\right). \end{align} $  (21) 
Considering (15), (19), (20), and (21), the timeevolution operator
$ \begin{array}{l} U(t)= \\ \frac{1}{2}\left[\!\! \begin{array}{ccc} u_1{u_4}^*+{u_1}^*u_4+u_2{u_3}^*+{u_2}^*u_3 & \left(u_2{u_3}^*u_1{u_4}^*+{u_1}^*u_4{u_2}^*u_3\right)i & \left(\ u^*_1u_3+{u_1u}^*_3u_2u^*_4+{u^*_2u}_4\right) \\ \left({u_1}^*u_4u_1{u_4}^*+u_2{u_3}^*{u_2}^*u_3\right)i & {u_1}^*u_4+u_1{u_4}^*u_2{u_3}^*{u_2}^*u_3 & \left({u_1u}^*_3\ u^*_1u_3u_2u^*_4+{u^*_2u}_4\right)i \\ 2(u_1{u_2}^*+{u_1}^*u_2) & 2({{u_1}^*u_2u}_1{u_2}^*)i & 2(u_1{u_1}^*{u_2}^*u_2) \end{array} \!\!\right]. \end{array} $  (22) 
In this paper, the desired gate is a Hadamard gate
$ \begin{align} {{ G}}_{\mathcal{H}}&=\ \frac{1}{\sqrt{2}}\big[\big(\big0\big \rangle+ \big1\big \rangle\big) \big \langle0\big+\big(\big0\big \rangle\big1\big \rangle \big) \big \langle1\big \big]\notag \\ &= \frac{1}{\sqrt{2}}\left[\begin{array}{cc} 1 & 1 \\ 1 &1 \end{array} \right]. \end{align} $  (23) 
To obtain the Hadamard gate
$ \begin{align} U_f=U_{f\mathcal{H}}=\begin{bmatrix} 0 & 0& 1 \\ 0 & 1& 0 \\ 1 & 0& 0 \end{bmatrix}.\end{align} $  (24) 
In Section Ⅱ, density matrix dynamics and desired quantum gate have been derived in the Bloch vector space, and we have obtained the dynamics of timeevolution operator
The Lyapunov stability theorem is used to determine the stability of a control system without need of solving the partial differential equations. It can also be used to design the control laws in order to obtain a stable control system. According to the Lyapunov stability theorem the dynamical system in (11), is stable if there is a scalar function
The Lyapunov function
$ \begin{align} \log(\mathcal{W}(t)) =&\, \left(\mathcal{W}(t)I\right) \frac{1}{2}({\left(\mathcal{W}(t)I\right)}^2 \nonumber\\ & +\frac{1}{3}({\left(\mathcal{W}(t)I\right)}^3\frac{1}{4} ({\left(\mathcal{W}(t)I\right)}^4+\cdots \end{align} $  (25) 
where
The first two terms of Mercator series in (25) are chosen, and the Lyapunov function in this paper is constructed by taking the square norm of two terms as
$ \begin{align} V(t)= {\left\\mathcal{L}(t)\right\}^2={\rm tr}({\mathcal{L}}^{\dagger }(t)\mathcal{L}(t)) \end{align} $  (26) 
in which,
Equation (26) asserts that,
To design the control laws, the first order time derivation of
$ \begin{align} \dot{V}(t)=&\ {\rm tr}\left(\frac{d}{dt}({\mathcal{L}}^{\dagger } (t)\mathcal{L}(t)\right)\nonumber\\ =&\ {\rm tr}\left(\dot{{\mathcal{L}}}^{\dagger} (t)\mathcal{L}(t)+{\mathcal{L}}^{\dagger } (t)\dot{\mathcal{L}}(t)\right) \end{align} $  (27) 
where
$ \begin{align} &\dot{\mathcal{L}}(t)=\dot{\mathcal{W}}(t) (\left(\ \mathcal{W}(t)I\right)\dot{\mathcal{W}}(t)) \nonumber\\ & {\mathcal{L}}^{\dagger }(t)={\mathcal{W}^{\dagger }(t)} I  \frac{\mathit{\boldsymbol{1}}}{\mathit{\boldsymbol{2}}} \mathit{\boldsymbol{\ }}{\mathit{\boldsymbol{(}}{\mathcal{W}^{\dagger }(t)} \mathit{\boldsymbol{}}I\mathit{\boldsymbol{)}}}^{\mathit{\boldsymbol{2}}} \nonumber\\ & \dot{{\mathcal{L}}}^{\dagger}(t)=\dot{{\mathcal{W}}}{\dagger }(t)^\left(\left({\mathcal{W}^{\dagger }(t)}I\right)\dot{{\mathcal{W}}}^{\dagger }(t)\right) \end{align} $  (28) 
in which,
By substituting
$ \begin{align} \dot{V}(t)=&\ {\rm tr}\Big[ \Big(\frac{1}{2}\dot{{U}^{\dagger}(t)}U_f{\mathcal{W}}^{\dagger }\nonumber\\ &\frac{1}{2}{\mathcal{W}}^{\dagger }\dot{{U}^{\dagger}(t)}U_f+2 \dot{{U}^{\dagger}(t)}U_f\Big)\mathcal{L}\nonumber\\ & +{\mathcal{L}}^{\dagger }(\frac{1}{2}{U_f}^{\dagger } \dot{U}(t)\mathcal{W}\frac{1}{2}\mathcal{W}{U_f}^{\dagger } \dot{U}(t)+2{U_f}^{\dagger }\dot{U}(t)\Big] \end{align} $  (29) 
where the first and second terms of the trace function are the conjugate transpose of each other. Moreover, all elements of the trace function are real matrices, so the trace of these two terms are equal, and (29) can be rewritten as
$ \begin{align} \dot{V}(t)=&\ 2{\rm tr}\Big[\Big(\frac{1}{2}\dot{{U^{\dagger }(t)}}U_f{\mathcal{W}}^{\dagger }\nonumber\\[1mm] &\frac{1}{2}{\mathcal{W}}^{\dagger }\dot{{U^{\dagger }(t)}} U_f+2\dot{{U^{\dagger }(t)}}U_f\Big)\mathcal{L}\Big]. \end{align} $  (30) 
Substituting the conjugate transpose of
$ \begin{align} \dot{V}(t)=&\ 2{\rm tr}\Big[\Big(\frac{1}{2}{U^{\dagger }(t)}{\left(A(t)+B\right)}^{\dagger }\ U_f{\mathcal{W}}^{\dagger }\nonumber\\ & \frac{1}{2}{\mathcal{W}}^{\dagger }{U^{\dagger }(t)}{\left(A(t)+B\right)}^{\dagger }\ U_f\nonumber\\ & +2{U^{\dagger }}(t){(A(t)+B)}^{\dagger }\ U_f\Big)\mathcal{L}\Big]. \end{align} $  (31) 
Substituting
$ \begin{align} \label{GrindEQ__32_} \dot{V}(t)=&\ 2{{\rm tr}}\Big[\Big(\frac{1}{2}{U^{\dagger }(t)}(f_x(t) A_x\nonumber\\ &+f_y(t)A_y+f_z(t)A_z +B)^ {\dagger }\ U_f{\mathcal{W}}^{\dagger }\nonumber\\ &\frac{1}{2}{\mathcal{W}}^{\dagger }{U^{\dagger }(t)} (f_x(t)A_x+f_y(t)A_y\\ &+f_z(t)A_z +B)^{\dagger }\ U_f +2{U^{\dagger }(t)}(f_x(t)A_x\nonumber\\ &+f_y(t)A_y+f_z(t)A_z +B)^{\dagger }\ U_f\Big)\mathcal{L}\Big] \end{align} $  (32) 
where
$ \begin{align} \dot{V}(t)=&\ f_x(t)2\ {\rm tr}\Big[\Big(\frac{1}{2}{U^{\dagger }(t)} A^\dagger_xU_f{\mathcal{W}}^{\dagger}\nonumber\\ &\frac{1}{2} {\mathcal{W}}^{\dagger}{U^{\dagger }(t)}A^\dagger_x U_f+2{U^{\dagger}(t)}A^\dagger_xU_f\Big) \mathcal{L}\Big]\nonumber\\ & +f_y(t)2\ {\rm tr}\Big[\Big(\frac{1}{2}{U(t)}^{\dagger} A^\dagger_yU_f{\mathcal{W}}^{\dagger }\nonumber\\ &\frac{1}{2} {\mathcal{W}}^{\dagger }{U^{\dagger }(t)}A^\dagger_y U_f+2{U}^{\dagger}(t)A^\dagger_yU_f\Big) \mathcal{L}\Big]\nonumber\\ & +f_z(t)2{\rm tr}\Big[\Big(\frac{1}{2}{U^{\dagger }(t)}A^\dagger_zU_f{\mathcal{W}}^{\dagger }\nonumber\\ &\frac{1}{2}{\mathcal{W}}^{\dagger} {U^{\dagger}(t)} A^\dagger_zU_f+2{U^{\dagger}(t)} A^\dagger_zU_f\Big)\mathcal{L}\Big]\nonumber\\ & +2{\rm tr}\Big[\Big(\frac{1}{2}{U^{\dagger }(t)}B^{\dagger }U_f{\mathcal{W}}^{\dagger}\nonumber\\ &\frac{1}{2}{\mathcal{W}}^{\dagger}{U^{\dagger}(t)}B^{\dagger} U_f+2{U^{\dagger}(t)}B^{\dagger}U_f\Big)\mathcal{L}\Big]. \end{align} $  (33) 
From (33) it is obvious that,
$ {\small \left(\dfrac{1}{2}{U(t)}^{\dagger }X^{\dagger }U_f{\mathcal{W}}^{\dagger }\dfrac{1}{2}{\mathcal{W}}^{\dagger }{U(t)}^{\dagger }X^{\dagger }U_f+2{U(t)}^{\dagger }X^{\dagger }U_f\right)\mathcal{L}} $ 
then these similar functions are defined as
$ \begin{align} S\left(X, t\right)=&\ 2{\rm tr}\Big[\Big(\frac{1}{2}{U(t)}^{\dagger } X^{\dagger }U_f{\mathcal{W}}^{\dagger }\nonumber\\ &\frac{1}{2}{\mathcal{W}}^{\dagger }{U(t)}^{\dagger }X^{\dagger }U_f+2{U(t)}^{\dagger }X^{\dagger }U_f\Big)\mathcal{L}\Big] \end{align} $  (34) 
in which
$ \begin{align} \dot{V}(t)=&\ f_x(t)S\left(A_x, t\right)+f_y(t)S\left(A_y, t\right)\nonumber\\ & +f_z(t)S\left(A_z, t\right)+S\left(B, t\right) \end{align} $  (35) 
while
Now the control task becomes to design the control functions
$ \begin{align} &f_x(t)=a_xS\left(A_x, t\right)h_x\frac{S\left(B, t\right)}{S\left(A_x, t\right)}\notag \\[1mm] & f_y(t)=a_yS\left(A_y, t\right)h_y\frac{S\left(B, t\right)}{S\left(A_y, t\right)} \notag \\[1mm] & f_z(t)=a_zS\left(A_z, t\right)h_z\frac{S\left(B, t\right)}{S\left(A_y, t\right)} \end{align} $  (36) 
where
Substituting (36) into (35), one gets
$ \begin{align} \dot{V}(t)=& a_x{S^2\left(A_x, t\right)}\nonumber\\ &a_y{S^2\left(A_y, t\right)} a_z{S^2\left(A_z, t\right)} \le 0 \end{align} $  (37) 
This means the control laws given by (36) can ensure
In this section, the control laws in (36) are used to prepare the Hadamard gate for a Markovian open quantum system, i.e., to drive the timeevolution operator
$ \begin{align} U_0=\begin{bmatrix} 1 & 0& 0 \\ 0 & 1& 0 \\ 0 & 0& 1 \end{bmatrix} \end{align} $  (38) 
$ \begin{align} U_f= \begin{bmatrix} 0 & 0& 1 \\ 0 & 1& 0 \\ 1 & 0& 0 \end{bmatrix}. \end{align} $  (39) 
Numerical simulations are conducted to investigate the performances of control laws and the dynamical behavior of the system. We mainly study the following three points:
1) The dynamics and characteristics of the timeevolution operator under the Lyapunovbased control are investigated. Meanwhile, the accuracy of preparation of the Hadamard gate is analyzed based on two performance indices: the fidelity F and the distance D, for different coupling strength
2) The effects of control laws on the control system performances are studied by analyzing the statetransfer from
3) The comparisons between different control methods are discussed.
A. Preparation of Hadamard Gate and Analysis of the Control Performance IndicesIn this subsection, the dynamics and characteristics of the timeevolution operator
In dynamical equation
$ U(t)=U_0+\frac{h}{6}\cdotp(K_1+2\cdotp K_2+2\cdotp K_3+K_4) $  (40) 
where
$ \begin{align} &K_1=\mathcal{F}\cdotp U_0\notag\\ &K_2=\mathcal{F}\cdotp(U_0+\frac{h}{2}\cdotp K_1)\notag\\ &K_3=\mathcal{F}\cdotp(U_0+\frac{h}{2}\cdotp K_2)\notag\\ &K_4=\mathcal{F}\cdotp(U_0+h\cdotp K_3) \end{align} $  (41) 
in which
$ \begin{align} \mathcal{F}=&\ (f_x(t)A_x+f_y(t)A_y+f_z(t)A_z+B)\nonumber\\ =&\ A(t)+B. \end{align} $  (42) 
In (40),
The fidelity and the distance are introduced to analyse the accuracy of quantum Hadamard gate preparation. The fidelity is defined as [27]
$ F=\frac{{\rm tr}\left(U(t)U^{\dagger }(t)\right)+{\left{\rm tr}({U_f}^{\dagger }U(t)\right}^2}{N(N+1)} $  (43) 
where
The distance is defined as
$ D={\left\U(t)U_f\right\}^2={\rm tr}({\left(U(t) U_f\right)}^{\dagger }\cdot \left(U(t)U_f\right)). $  (44) 
Accordingly, the distance gives the perception whether
$ D<{10}^{4} $  (45) 
which is the distance criterion for valid operator preparations.
As the system is an open quantum system, when the coupling strength
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Fig. 1 The fidelity under control laws for the AD system when 
One can see from Fig. 1 that, when
Fig. 2 is the result of the distance when preparing the Hadamard gate for the AD Markovian open quantum system with
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Fig. 2 The distance under control laws for the AD system when (a) 
The function of control laws consists of two parts: the first is the preparation, and the second is the preservation. During the preparation part, the desired gate is prepared, and two control performance indices, i.e., density and fidelity, tend to reach the minimum and maximum values, respectively. In the preservation part, the desired gate remains stable under the action of the control laws. The effects of control laws in the preservation part eliminate the dissipation of the system which emerges as the fluctuations.
Table Ⅰ is the parameters in (36) selected in experiments in order to have the maximum fidelity and the minimum distance in the shortest possible time. The control laws as the function of time with
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Fig. 3 Control laws as the function of the time when 
In this subsection, in order to study the relation between the density matrix and the gate, the numerical simulation of corresponding statetransfer from the arbitrary identity matrix
Let the initial vector be
$ \begin{align} r_f= U_f\cdotp r_0 = \begin{bmatrix} 0 & 0& 1 \\ 0 & 1& 0 \\ 1 & 0& 0 \end{bmatrix} \cdot\left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array} \right]=\left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array} \right]. \end{align} $  (46) 
To find out the corresponding density matrix, the initial vector
$ \begin{align} {\rho }_0=\left[\begin{array}{cc} 0.5 & 0.5 \\ 0.5 & 0.5 \end{array} \right], \quad {\rho }_f=\left[ \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right]. \end{align} $  (47) 
Fig. 4 illustrates the trajectory of the timeevolution density matrix as a function of time for the AD Markovian open quantum system under the designed control laws.
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Fig. 4 Statetransfer from 
Based on the principle of Von Neumann, the diagonal elements of a density matrix can be interpreted as the probability. The trace of a density matrix must be normalized, which means the sum of the diagonal elements of timeevolution density matrix, i.e.,
In [30], the optimal control theory is applied to a twolevel open quantum system to prepare the Hadamard gate by minimizing an energytype cost functional. 25 a.u. time was used and the performance of
In [18], the Lyapunov control method is used to prepare a Not gate for a twolevel open quantum system. The performance of
This paper has prepared a Hadamard gate for the twolevel AD Markovian open quantum system based on the Lyapunov stability theorem. The controlled system dynamics are obtained in the Bloch vector representation. Two control performance indices, i.e, the fidelity and the distance are investigated, and numerical simulations are implemented under the MATLAB environment with different coupling strength
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