# 物理代写|热力学代写thermodynamics代考|MEC302

## 物理代写|热力学代写thermodynamics代考|Optimized Control of Transfer between Multipartite Open-System

We resort to the general BOTOC approach (detailed in $\mathrm{Ch} .12$ ) to the control of arbitrary quantum operations in multidimensional open systems. The desired operation is disturbed via operators $\hat{S} \otimes \hat{B}$, where $\hat{S}$ is the traceless operator of the system, which is hereafter assumed to consist of qubits, and $\hat{B}$ is the bath operator. One can choose controls to maximize the operation fidelity by acting on the multiqubit system according to the following protocol, which holds to second order in the system-bath coupling:
(i) The control (modulation) transforms the system-bath coupling operators to the time-dependent form $\hat{S}(t) \otimes \hat{B}(t)$ in the interaction picture, via the set of time-dependent coefficients $\epsilon_i(t)$ that define a notation in the Pauli basis $\hat{\sigma}_i$,

$$\hat{S}(t)=\sum_i \epsilon_i(t) \hat{\sigma}i .$$ (ii) We next write the time-independent gradient-control matrix (12.31), describing how the fidelity score changes for each pair of the Paui basis operators, $$\Xi{i j} \equiv \overline{\left\langle\psi\left|\left[\sigma_i, \sigma_j|\psi\rangle\langle\psi|\right]\right| \psi\right\rangle},$$
the overline being an average over all possible initial states.
(iii) Using the matrix $\Xi$ whose elements are $\Xi_{i j}$, one arrives at the following expression for the average fidelity of the desired operation (to second-order accuracy in the system-bath coupling):
$$\bar{f}(t)=1-t \int_{-\infty}^{\infty} d \omega \operatorname{Tr}\left[\boldsymbol{G}(\omega) \boldsymbol{F}t(\omega)\right],$$ where $\boldsymbol{G}(\omega)$ is the bath-coupling (-response) spectral matrix defined in (12.32) and the modulation (control) spectral matrix $\boldsymbol{F}_t(\omega)$ is defined in (12.37) according to the operation, via the gradient-control matrix $\Xi{i j}$.
(iv) The fidelity is maximized by the variational Euler-Lagrange method described in Chapter 12 that minimizes the overlap between $\boldsymbol{G}(\omega)$ and $\boldsymbol{F}_t(\omega)$ under the constraint of a given control energy.

## 物理代写|热力学代写thermodynamics代考|Optimized State Transfer from Noisy to Quiet Qubits

The general approach outlined above will next be used to optimize the fidelityversus-speed trade-off in the transfer of quantum states from a fragile (noisy) qubit to a robust (quiescent) qubit. We focus here on the case of two resonant qubits with temporally controlled coupling strength. The free Hamiltonian is then
\begin{aligned} &\hat{H}{\mathrm{S}}(t)=\frac{\hbar \omega_0}{2}\left(\hat{\sigma}{z 1}+\hat{\sigma}{z 2}\right)+\hat{H}{\mathrm{c}}(t), \ &\hat{H}{\mathrm{c}}(t)=\hbar V(t) \hat{\sigma}{x 1} \otimes \hat{\sigma}{x 2} \end{aligned} where $\hat{H}{\mathrm{c}}(t)$ is the two-qubit controlled-interaction Hamiltonian, $V(t)$ being the interaction amplitude (see Fig. 14.1-inset), where the interaction amplitude is adjustable by an external laser field. The system-bath interaction Hamiltonian is taken to represent proper dephasing in the noisy source qubit 1 due to the bath operator $\hat{B}$, whereas the target qubit 2 is quiescent, that is, robust against decoherence,
$$\hat{H}{\mathrm{I}}=\hbar \hat{S} \otimes \hat{B}(t)=\hbar \hat{\sigma}{z 1} \otimes \hat{B}(t),$$ where $\hat{B}(t)$ is the bath operator $\hat{B}$ that evolves under the action of the free bath Hamiltonian $\hat{H}_{\mathrm{B}}$. This model can be generalized to allow for any asymmetry between the dephasing (decoherence) of the two qubits.

## 物理代写|热力学代写热力学代考|多部开放系统间传输的优化控制

(i)控制(调制)将系统浴耦合运算符转换为交互图中与时间相关的形式$\hat{S}(t) \otimes \hat{B}(t)$，通过一组与时间相关的系数$\epsilon_i(t)$，该系数定义了泡利基$\hat{\sigma}_i$中的符号，

$$\hat{S}(t)=\sum_i \epsilon_i(t) \hat{\sigma}i .$$ (ii)接下来，我们编写了时间无关的梯度控制矩阵(12.31)，描述了每一对Paui基算子的保真度评分是如何变化的， $$\Xi{i j} \equiv \overline{\left\langle\psi\left|\left[\sigma_i, \sigma_j|\psi\rangle\langle\psi|\right]\right| \psi\right\rangle},$$

(iii)使用矩阵 $\Xi$ 它的元素是 $\Xi_{i j}$，就会得到以下表达式来表示所需操作的平均保真度(到系统浴耦合中的二阶精度):
$$\bar{f}(t)=1-t \int_{-\infty}^{\infty} d \omega \operatorname{Tr}\left[\boldsymbol{G}(\omega) \boldsymbol{F}t(\omega)\right],$$ 哪里 $\boldsymbol{G}(\omega)$ (12.32)中是否定义了浴耦合(-响应)光谱矩阵和调制(控制)光谱矩阵 $\boldsymbol{F}_t(\omega)$ 在(12.37)中根据操作定义，通过梯度控制矩阵 $\Xi{i j}$
(iv)通过第12章描述的变分欧拉-拉格朗日方法来最大化保真度 $\boldsymbol{G}(\omega)$ 和 $\boldsymbol{F}_t(\omega)$ 在给定控制能量的约束下。

## 物理代写|热力学代写热力学代考|从噪声量子位到安静量子位的优化状态传输

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\begin{aligned} &\hat{H}{\mathrm{S}}(t)=\frac{\hbar \omega_0}{2}\left(\hat{\sigma}{z 1}+\hat{\sigma}{z 2}\right)+\hat{H}{\mathrm{c}}(t), \ &\hat{H}{\mathrm{c}}(t)=\hbar V(t) \hat{\sigma}{x 1} \otimes \hat{\sigma}{x 2} \end{aligned}，其中$\hat{H}{\mathrm{c}}(t)$为双量子位控制的相互作用哈密顿量，$V(t)$为相互作用振幅(见图14.1-插图)，其中相互作用振幅可由外部激光场调节。系统-浴相互作用哈密顿量被用来表示噪声源量子比特1由于浴算符$\hat{B}$的适当失相，而目标量子比特2是静止的，即对退相干具有鲁棒性，
$$\hat{H}{\mathrm{I}}=\hbar \hat{S} \otimes \hat{B}(t)=\hbar \hat{\sigma}{z 1} \otimes \hat{B}(t),$$，其中$\hat{B}(t)$是浴算符$\hat{B}$，在自由浴哈密顿量$\hat{H}_{\mathrm{B}}$的作用下进化。该模型可以推广到允许两个量子位的失相(退相干)之间的任何不对称

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