Fidelity optimization for holonomic quantum gates in dissipative environments

We analyze the performance of holonomic quantum gates in semiconductor quantum dots, driven by ultrafast lasers, under the effect of a dissipative environment. The environment is modeled as a thermal bath of oscillators linearly coupled with the electron states of the quantum dot. Standard techniques make the problem amenable to a numerical treatment and allow one to determine the fidelity as a function of all the relevant physical parameters. As a consequence of our analysis, we show that the disturbance of the environment can be (approximately) suppressed and the performance of the gate optimized—provided that the thermal bath is purely super-Ohmic. We conclude by showing that such an optimization is impossible for Ohmic environments.

Figure 1

Fidelity $\mathcal{F}$ for gate $1$ as a function of temperature for different $k_3$ [expressed in $(\mathrm{meV}{)}^{-2}$] in the presence of a super-Ohmic environment. The points are the computer simulation results, and the curves have the form $\mathcal{F}=1-{\sum }_{j=\pm }{\eta }_j{\Gamma }^j{t}_{ad}$ (${\eta }_{-}=3\times {10}^{-2}$ and ${\eta }_{+}=0.7$). Inset: fidelity $\mathcal{F}$ as a function of $k_3$ for two different temperatures ($T_1∕\Omega =1.6\times {10}^{-2}$ and $T_2∕\Omega =5\times {10}^{-3}$). Parameters: $ϵ=1\mathrm{eV}$, $\Omega =25\mathrm{meV}$, $t_{ad}=7.5\mathrm{ps}$, and ${\omega }_c=0.5\mathrm{meV}$.

Figure 2

Fidelity $\mathcal{F}$ for gate $1$ subject to super-Ohmic spectral density as a function of $t_{ad}$ [$T∕\Omega =8.5\times {10}^{-3}$, $k_3={10}^{-1}$ $(\mathrm{meV}{)}^{-2}$, and $\Omega t_{ad}=280$]. The adiabatic time is normalized to $5\mathrm{ps}$. Parameters as in Fig. 1 except $t_{ad}$.

Figure 3

Fidelity $\mathcal{F}$ for gate $1$ subject to super-Ohmic (dashed line), Ohmic (dotted line), and both (solid line) spectral densities as a function of $T∕\Omega$. Inset: fidelity $\mathcal{F}$ for super-Ohmic plus Ohmic environment as a function of $t_{ad}$ with $\Omega t_{ad}=280$ [$k_1=4\times {10}^{-4}$ and $k_3={10}^{-1}(\mathrm{meV}{)}^{-2}$]. Parameters as in Fig. 1.

Figure 4

Fidelity for two-qubit gate subject to super-Ohmic and Ohmic spectral density as a function of $T∕\Omega$ for $k_1={10}^{-4}$ and $k_3={10}^{-2}(\mathrm{meV}{)}^{-2}$. Parameters as in Fig. 1 except $t_{ad}=0.8\mathrm{ns}$ and $\Omega =0.2\mathrm{meV}$.