# 物理代写|热力学代写thermodynamics代考|Optimal Gate Protection

## 物理代写|热力学代写thermodynamics代考|Optimal Gate Protection

The protection of a given quantum operation from decoherence is most effective under bath-optimized task-oriented control (BOTOC), expounded in Chapter 12. Here, we consider the implementation of a quantum-gate unitary operation within a given “gate time” $t$ for a pure input state $|\Psi\rangle$. In the interaction picture with respect to the desired gate operation, the projector $\hat{P}=\hat{\varrho}(0)=|\Psi\rangle\langle\Psi|$ is then used as the gradient operator, so that (12.22) is satisfied. Then, (12.24) yields the fidelity change as the score
$$P=\langle\Psi|\Delta \hat{\varrho}| \Psi\rangle=-t^2\left\langle\left\langle\Psi\left|\hat{H}^2\right| \Psi\right\rangle-\langle\Psi|\hat{H}| \Psi\rangle^2\right\rangle_{\mathrm{B}}$$
To eliminate the dependence on $|\Psi\rangle$, we uniformly average over all $|\Psi\rangle$, whereby for any two operators $\hat{A}$ and $\hat{B}$ :
$$\overline{\langle\Psi|\hat{A}| \Psi\rangle\langle\Psi|\hat{B}| \Psi\rangle}=\frac{\operatorname{Tr} \hat{A} \hat{B}+\operatorname{Tr} \hat{A} \operatorname{Tr} \hat{B}}{d(d+1)},$$
$d$ being the Hilbert-space dimensionality of the system. The average score is then
$$\bar{P}=-t^2 \frac{d}{d+1}\left\langle\hat{H}^2\right\rangle_{\mathrm{id}}$$
where [in the notation of (12.6)] $\langle\ldots\rangle_{\text {id }}=\operatorname{Tr}\left(d^{-1} \hat{I} \otimes \hat{\varrho}B \ldots\right)$. Here we have used $\operatorname{Tr}{\mathrm{S}} \hat{H}=0$, corresponding to $\operatorname{Tr} \hat{S}j=0$. On account of $(12.21),\langle\hat{H}\rangle{\mathrm{B}}=\left\langle\hat{H}{\mathrm{I}}\right\rangle{\mathrm{B}}=0$, $\langle\hat{H}\rangle_{\text {id }}=0$, so that (13.21) is proportional to the variance of the Hamiltonian: $\operatorname{Var}(\hat{H})=\left\langle\hat{H}^2\right\rangle_{\text {id }}-\langle\hat{H}\rangle_{\text {id }}^2$. The gate error $\mathcal{E}$, which is the average fidelity decline (or the infidelity), then satisfies
$$\mathcal{E} \equiv-\bar{P}=t^2 \frac{d}{d+1} \operatorname{Var}(\hat{H})$$
The average over the initial states in the matrix $\Xi$, defined in (12.31), yields $\bar{\Xi}=$ $-\frac{d}{d+1} \boldsymbol{I}$, upon using $\operatorname{Tr}\left(\hat{S}j \hat{S}_k\right)=d \delta{j k}$ and $\operatorname{Tr} \hat{S}j=0$. Hence $$\mathcal{E}=\frac{d}{d+1} \int{-\infty}^{\infty} d \omega \operatorname{Tr}\left[\boldsymbol{\epsilon}t(\omega) \boldsymbol{\epsilon}_t^{\dagger}(\omega) \boldsymbol{G}(\omega)\right]$$ where $\boldsymbol{G}(\omega)$ and $\boldsymbol{\epsilon}_t(\omega)$ are given by (12.32) and (12.33), respectively. Because of the requirement that $\mathcal{E} \geq 0, \boldsymbol{G}(\omega)$ must be a positive semidefinite matrix for any $\omega$. BOTOC then aims at finding the evolution operator of the system, $\hat{U}(t)$ (as in the control examples in Sec. 12.2), that minimizes $\mathcal{E}$, subject to the condition that the desired gate is executed over time interval $t{\mathrm{f}}$.

We may therefore conclude that, whereas each gate operation should be as fast as possible, the optimal overall pulse sequence may take longer than the gate time because of the storage control duration. This general principle is unparalleled by other approaches.

## 物理代写|热力学代写thermodynamics代考|Implementation in Trapped-Ion Systems

A possible implementation of this approach may involve a string of ions in a linear trap. The qubits, encoded by two internal states of each ion $\left[|g(e)\rangle_j\right]$, are manipulated by laser beams. An additional qubit, encoded by the ground and first excited common vibrational levels $\left(|0(1)\rangle_N\right)$, acts as the “bus mode.” The qubit gates are executed by applying laser pulses on the “carrier” $\left[\Omega_j^{(1)}(t),|g\rangle \leftrightarrow|e\rangle\right]$, the “blue sideband” $\left[\Omega_{j N}^{(2) \Phi}(t),|g\rangle|0\rangle \leftrightarrow|e\rangle|1\rangle\right]$, and the “red sideband” $\left[\Omega_{j N}^{(2) \omega}(t),|g\rangle|1\rangle \leftrightarrow\right.$ $|e\rangle|0\rangle]$ of the electronic quadrupole transition [Fig. 13.1(a)]. In a harmonic trap, the blue sideband also couples to higher levels, for example, $|g\rangle|1\rangle \leftrightarrow|e\rangle|2\rangle$, and the red one to $|e\rangle|1\rangle \leftrightarrow|g\rangle|2\rangle$. Such unwarranted excitations complicate and hamper the fidelity of the concurrent application of both two-qubit gates. These excitations may be suppressed by resorting to trap anharmonicity. Dephasing in the trapped-ion system arises due to magnetic-field fluctuations that give rise to random Zeeman shifts of the qubit levels. Simulations of a SWAP gate involving the lowest two common vibrational levels (in an anharmonic trap) in the presence of dephasing [Fig. 13.1(b)] show that the gate fidelity may be raised by means of the optimized pulse sequence compared to its standard counterpart, despite the longer duration of the former.

# 热力学代考

## 物理代写|热力学代写thermodynamics代考|Optimal Gate Protection

$$P=\langle\Psi|\Delta \hat{\varrho}| \Psi\rangle=-t^2\left\langle\left\langle\Psi\left|\hat{H}^2\right| \Psi\right\rangle-\langle\Psi|\hat{H}| \Psi\rangle^2\right\rangle_{\mathrm{B}}$$

$$\overline{\langle\Psi|\hat{A}| \Psi\rangle\langle\Psi|\hat{B}| \Psi\rangle}=\frac{\operatorname{Tr} \hat{A} \hat{B}+\operatorname{Tr} \hat{A} \operatorname{Tr} \hat{B}}{d(d+1)},$$
$d$ 是系统的希尔伯特空间维数。那么平均分就是
$$\bar{P}=-t^2 \frac{d}{d+1}\left\langle\hat{H}^2\right\rangle_{\text {id }}$$

$$\mathcal{E} \equiv-\bar{P}=t^2 \frac{d}{d+1} \operatorname{Var}(\hat{H})$$

$$\mathcal{E}=\frac{d}{d+1} \int-\infty^{\infty} d \omega \operatorname{Tr}\left[\boldsymbol{\epsilon t}(\omega) \boldsymbol{\epsilon}_t^{\dagger}(\omega) \boldsymbol{G}(\omega)\right]$$

## 物理代写|热力学代写thermodynamics代考|Implementation in Trapped-Ion Systems

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