# 物理代写|热力学代写thermodynamics代考|MECH3720

## 物理代写|热力学代写thermodynamics代考|Optimized Control of State Transfer through Noisy

QI processing and communication require reliable transfer of QI between distant nodes of a network, despite the vulnerability of the process to noise and perturbations. A simple control involves only the source and target (boundary) qubits that are weakly coupled to a chain of spins with identical couplings. Quantum states can then be transmitted with arbitrarily high fidelity at the expense of increasing the transfer time. Yet, even if identical couplings could be ensured, the required slowdown of the transfer would be detrimental, because of omnipresent decoherence.

To overcome this problem, we may optimize the trade-off between fidelity and transfer speed through noisy spin channels. This approach employs temporal modulation of the couplings between the boundary qubits and the channel spins. This modulation is treated as dynamical control of the boundary qubits that are coupled to a fermionic bath that is the source of noise. This modulation aims at realizing an optimal spectral filter that blocks transfer via the channel eigenmodes that are responsible for noise-induced leakage of the QI. We show that under optimal modulation, the fidelity and the speed of transfer can be improved by several orders of magnitude, and the fastest possible transfer is achievable (for a given fidelity).
Let us specifically consider a chain of $N+2$ spin- $1 / 2$ particles with $\mathrm{XX}$ interactions between nearest neighbors. The chain and boundary-coupling Hamiltonians, $H_0$ and $H_{\mathrm{bc}}$, are given by
\begin{aligned} H_0 &=\hbar \sum_{i=1}^{N-1} \frac{J_i}{2}\left(\sigma_{x i} \sigma_{x, i+1}+\sigma_{y, i} \sigma_{y, i+1}\right) \ H_{\mathrm{bc}}(t) &=\hbar g(t) \sum_{i \in{0, N}} \frac{J_i}{2}\left(\sigma_{x i} \sigma_{x, i+1}+\sigma_{y, i} \sigma_{y, i+1}\right) \end{aligned}
Here $\sigma_{x(y) i}$ are the appropriate Pauli matrices, $J_i$ are the corresponding exchangeinteraction couplings, and $g(t)$ is the temporally modulated coupling strength. The magnetization-conserving $H_0$ can be mapped onto a noninteracting fermionic Hamiltonian that has the diagonal, particle-conserving form $H_0=\sum_{k=1}^N \hbar \omega_k b_k^{\dagger} b_k$, where $b_k^{\dagger}$ populates a fermionic single-particle eigenstate $\left|\omega_k\right\rangle$ of energy $\omega_k$.

## 物理代写|热力学代写thermodynamics代考|Optimal Filter Design

We resort to modulation as a tool to minimize the infidelity $\zeta\left(t_{\mathrm{f}}\right)=1-f_{0, N+1}\left(t_{\mathrm{f}}\right)$ by rendering the overlap between the bath and system spectra as small as possible,
$$\min \zeta\left(t_{\mathrm{f}}\right)=\min \operatorname{Re} \int_0^{t_f} d t \int_0^t d t^{\prime} \sum_{\pm} \Omega_{\pm}(t) \Omega_{\pm}\left(t^{\prime}\right) \Phi_{\pm}\left(t-t^{\prime}\right) .$$
Here $\Phi_{\pm}(t)=\sum_{k \in k \text { odd(even) }}\left|\eta_k\right|^2 e^{-i \omega_k t}$ are the correlation functions associated with the odd (even) bath modes, respectively. The corresponding dynamicalcontrol functions are $\Omega_{+}(t)=g(t) \cos [\sqrt{2} \phi(t)]$ and $\Omega_{-}(t)=g(t)$. In the energy domain, (14.28) has the form
$$\zeta\left(t_{\mathrm{f}}\right)=\int_{-\infty}^{\infty} d \omega \sum_{\pm} F_{t f, \pm}(\omega) G_{\pm}(\omega) .$$
Here,
$$G_{\pm}(\omega)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} d t \Phi_{\pm}(t) e^{i \omega t}$$
are the bath-spectrum functions, corresponding to odd (even) parity modes, and
$$F_{t_i, \pm}(\omega)=\frac{1}{2 \pi}\left|\int_0^{t_f} d t \Omega_{\pm}(t) e^{i \omega t}\right|^2$$
are the filter-spectrum functions, which can be designed by the modulation control. The optimal modulation minimizes the overlap integrals of $G_{\pm}(\omega)$ and $F_{l_1, \pm}(\omega)$ for a given $t_{\mathrm{f}}$ by the variational Euler-Lagrange (EL) method.

We assume the channel to be symmetric with respect to the source and target qubits and the number of eigenvalues to be odd (Fig. 14.3 – top inset). Under these assumptions, the central eigenvalue is invariant under noise on the couplings, provided a gap exists between this eigenvalue and the adjacent ones, that is, they are not strongly mixed by the noise, so as not to overlap. We also assume that the discreteness of the bath spectrum of the quantum channel is smoothed out by the noise. Then, considering the central eigenvalue as part of the system spectrum, a common characteristic of $G_{\pm}(\omega)$ is to have a central gap [Fig. 14.3(a)].

## 物理代写|热力学代写热力学代考|通过噪声的状态转移的优化控制

QI处理和通信需要在网络的遥远节点之间进行可靠的QI传输，尽管该过程容易受到噪声和扰动的影响。一个简单的控制只涉及源和目标(边界)量子位，它们弱耦合到具有相同耦合的自旋链。量子态可以以任意高保真度传输，代价是增加传输时间。然而，即使能够保证相同的耦合，由于无所不在的退相干，所需的传输减缓也将是有害的

\begin{aligned} H_0 &=\hbar \sum_{i=1}^{N-1} \frac{J_i}{2}\left(\sigma_{x i} \sigma_{x, i+1}+\sigma_{y, i} \sigma_{y, i+1}\right) \ H_{\mathrm{bc}}(t) &=\hbar g(t) \sum_{i \in{0, N}} \frac{J_i}{2}\left(\sigma_{x i} \sigma_{x, i+1}+\sigma_{y, i} \sigma_{y, i+1}\right) \end{aligned}

## 物理代写|热力学代写热力学代考|最优滤波器设计

$$\min \zeta\left(t_{\mathrm{f}}\right)=\min \operatorname{Re} \int_0^{t_f} d t \int_0^t d t^{\prime} \sum_{\pm} \Omega_{\pm}(t) \Omega_{\pm}\left(t^{\prime}\right) \Phi_{\pm}\left(t-t^{\prime}\right) .$$

$$\zeta\left(t_{\mathrm{f}}\right)=\int_{-\infty}^{\infty} d \omega \sum_{\pm} F_{t f, \pm}(\omega) G_{\pm}(\omega) .$$

$$G_{\pm}(\omega)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} d t \Phi_{\pm}(t) e^{i \omega t}$$

$$F_{t_i, \pm}(\omega)=\frac{1}{2 \pi}\left|\int_0^{t_f} d t \Omega_{\pm}(t) e^{i \omega t}\right|^2$$

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