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## 物理代写|热力学代写thermodynamics代考|Multiqubit Dynamical Symmetry Control

Below, we mostly assume that the particles are qubits [Fig. 13.4(a)], so that they have only one excited state $|1\rangle_j \equiv|e\rangle_j$. The bath-coupling off-diagonal matrix element $G_{j j^{\prime}}(\omega)$ is then associated with the interaction of qubits $j$ and $j^{\prime}$ via the bath, and $\tilde{F}{t, j j^{\prime}}(\omega)$ is the dynamical modulation matrix, which we design at will to suppress the decoherence or the relaxation. The relaxation or decoherence of the system of qubits is described by the decoherence matrix (13.33) in the simplified notation $J{j j^{\prime}}(t)$.

The diagonal elements, $J_{j j}(t)$, are determined by the coupling of individual qubits (labeled by $j$ ) to their respective baths. On the other hand, the off-diagonal elements, $J_{j j^{\prime}}(t)$ with $j \neq j^{\prime}$, represent the results of cross-decoherence induced on the $j$ th qubit by the $j^{\prime}$ th qubit. Such cross-decoherence exists only for qubits coupled to a common bath mode or sharing a correlated dephasing-noise mode [Fig. 13.4(a)]. Specifically, cross-dephasing stems from the cross correlation of the stochastic noise functions of qubits $j$ and $j^{\prime}$. Thus, for example, ions subject to the same fluctuating magnetic field undergo cross-dephasing. This off-diagonal cross-decoherence may be viewed as either virtual off-resonant or real (resonant) emission of a quantum into the bath by qubit $j$ and its reabsorption by qubit $j^{\prime}$ [Fig. 13.4(a)].

In the relaxation scenario, the modulation spectral function $\tilde{F}_{r, j j^{\prime}}(\omega)$ represents the effective change in qubit $j$ ‘s energy splitting or AC Stark shift [Fig. 13.4(b)]. In the dephasing scenario, it determines the time-dependent Rabi splitting of the I $\pm$ ) states [Fig. 13.4(c)]. In both scenarios, the spectral function in the integrand in (13.33) represents the effective coupling between the qubits via the bath, filtered by the dynamical control (modulation). This filter determines which spectral bath modes the qubits are coupled to.

## 物理代写|热力学代写thermodynamics代考|Multiqubit Decoherence Control at T

Our system is comprised of $N$ identical qubits labeled $j=1, \ldots, N$ with ground and excited states $|g\rangle_j$ and $|e\rangle_j$, respectively, and energy separation $\hbar \omega_{\mathrm{a}}$ (Fig. 13.6). Each qubit may be differently (weakly) coupled to a bath, the Hamiltonian being (henceforth we take $\hbar=1$ )
$$H(t)=H_{\mathrm{S}}+H_{\mathrm{B}}+H_{\mathrm{I}}+H_{\mathrm{C}}(t)$$

$$H_{\mathrm{S}}=\omega_{\mathrm{a}} \sum_{j=1}^N|e\rangle_{j j}\langle e|, \quad H_{\mathrm{I}}=\sum_{j=1}^N \hat{S}j \hat{B}_j .$$ Here $H{\mathrm{S}}, H_{\mathrm{B}}$, and $H_{\mathrm{I}}$ are the system, bath, and interaction Hamiltonians, respectively, $H_C(t)$ is the time-dependent control Hamiltonian, $\hat{B}_j$ being the $j$ th qubit bath operator, and $\hat{S}_j$ the system-bath coupling operator of qubit $j$.

Two kinds of decoherence can be similarly treated: (i) Relaxation, caused by an off-diagonal system-bath coupling Hamiltonian (in the energy basis), corresponding to the $X$-Pauli matrix of the $j$ th qubit,
$$\hat{s}j=\sigma{x j}=|e\rangle_{j j}\langle g|+| g\rangle_{j j}\langle e| .$$
Relaxation involves temperature-dependent level-population change, as well as the erasure (randomization) of the phase relations. Such a process is controllable if each qubit is subject to a strong but off-resonant driving field that causes an $A C$ Stark shift [Fig. 13.4(b)], that is, energy modulation of the system Hamiltonian,
$$H_{\mathrm{C}}(t)=\sum_{j=1}^N \Delta_j(t)|e\rangle_{j j}\langle e| .$$
(ii) Dephasing is caused by a system-bath coupling Hamiltonian that is diagonal in the energy basis, via the Z-Pauli matrix of the $j$ th qubit,
$$\hat{S}j=\sigma{z j}=|e\rangle_{j j}\langle e|-| g\rangle_{j j}\langle g| .$$

# 热力学代写

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## 物理代写|热力学代写热力学代考|多量子比特退相干控制at T

$$H(t)=H_{\mathrm{S}}+H_{\mathrm{B}}+H_{\mathrm{I}}+H_{\mathrm{C}}(t)$$

$$H_{\mathrm{S}}=\omega_{\mathrm{a}} \sum_{j=1}^N|e\rangle_{j j}\langle e|, \quad H_{\mathrm{I}}=\sum_{j=1}^N \hat{S}j \hat{B}_j .$$这里$H{\mathrm{S}}, H_{\mathrm{B}}$和$H_{\mathrm{I}}$分别是系统、浴和交互哈密顿量，$H_C(t)$是时间依赖的控制哈密顿量，$\hat{B}_j$是$j$第th量子比特浴算符，$\hat{S}_j$是量子比特$j$的系统浴耦合算符 两种退相干可以类似地处理:(i)弛豫，由非对角系统-浴耦合哈密顿量引起(在能量基础上)，对应于$j$第th量子比特的$X$ -泡利矩阵，
$$\hat{s}j=\sigma{x j}=|e\rangle_{j j}\langle g|+| g\rangle_{j j}\langle e| .$$

$$H_{\mathrm{C}}(t)=\sum_{j=1}^N \Delta_j(t)|e\rangle_{j j}\langle e| .$$
(ii)失相是由一个在能量基础上是对角的系统-槽耦合哈密顿量引起的，通过第$j$个量子比特的z -泡利矩阵，
$$\hat{S}j=\sigma{z j}=|e\rangle_{j j}\langle e|-| g\rangle_{j j}\langle g| .$$

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## MATLAB代写

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