Universal quantum control over Majorana zero modes

Abstract

Majorana zero mode (MZM) exhibits inherent resilience to local parametric fluctuations, due to the topological protection mechanism in the non-Abelian braiding statistics of the anyonic quasiparticles. In this paper, we construct the braiding operations between an arbitrary pair in three MZMs under the theoretical framework of universal quantum control. Largely detuned driving fields on the mediator, a local defect of lattice, enable indirect and tunable exchange interaction between arbitrary two MZMs. The braiding operations can then emerge through the time evolution along the universal nonadiabatic passages, whose robustness against driving-field errors and quasiparticle poisoning can be substantially enhanced by the rapid modulation over the passage-dependent global phase. Moreover, a chiral transfer for population on MZMs along the universal passage can be perfectly demonstrated in both clockwise and counterclockwise manners without eliminating the mediator defect. Our protocol is linear scalable and provides an avenue towards the universal quantum control over MZMs, which is fundamental and essential for topological quantum computation.

Figures (6)

• Figure 1: (a) Sketch of an atomic system comprising a lattice defect and three one-dimensional $N$-site Kitaev chains. LD can be coupled to MZMs $\gamma_{1,1}$, $\gamma_{1,2}$, and $\gamma_{1,3}$ through the local off-resonant driving fields. Inset: the four-level transition diagram for the effective subspace of LD and MZMs. (b) Effective two-level transition diagram for the braiding operation between two MZMs $\gamma_{1,j}$ and $\gamma_{1,k}$, $j\ne k\in{1,2,3}$.
• Figure 2: Fidelity $\mathcal{F}(\theta)$ for the states $|n\rangle$, $n=\gamma_{1,j},\gamma_{1,k},\gamma_{1,l},e$, as a function of the parameter $\theta(t)$ during the braiding process with the initial population in MZMs (a) $\gamma_{1,j}$ and (b) $\gamma_{1,k}$. $\Delta_j(t)=\Delta_0-\Delta(t)/2$ and $\Delta_k(t)=\Delta_0+\Delta(t)/2$ where $\Delta(t)$ in Eq. (\ref{['Deltar']}) and it is in the order of $\Delta(t)\sim10^{-2}\Delta_0$. $\Delta_l=2\Delta_0$. Under $\Omega_1(t)=\Omega_2(t)=\Omega_3(t)$, the effective Rabi frequency $\Omega(t)$ satisfies Eqs. (\ref{['EffPara']}) and (\ref{['Om0']}) with $\theta(t)=\pi t/(2T)$ ($T$ is the control period) and $f(\theta,\alpha)=0$. The Rabi frequency is in the order of $\Omega_1(t)\sim 10^{-2}\Delta_0$.
• Figure 3: Fidelity $\mathcal{F}[\theta(T)=\pi/2]$ about the target state $|\gamma_{1,k}\rangle$ versus the perturbative coefficient $\epsilon$, controlled by the full Hamiltonian $H_{\rm tot}(t)$ (\ref{['HamDri']}) with (a) deviated transition frequency $\omega$ in Eq. (\ref{['HamDriLocal']}) and (b) deviated Rabi frequency $\Omega_1(t)$ in Eq. (\ref{['HamDriGlob']}), under the global phase $f(\theta)$ in Eq. (\ref{['phasef']}) with various coefficients $\lambda$. The other parameters are the same as Fig. \ref{['Braidp']}(a).
• Figure 4: Fidelity $\mathcal{F}[\theta(T)=\pi/2]$ about the target state $|\gamma_{1,k}\rangle$ versus the perturbative coefficient $\epsilon$, controlled by the nonideal Hamiltonian $\tilde{H}_{\rm tot}(t)$ in Eq. (\ref{['Hampoison']}). The other parameters are the same as Fig. \ref{['Braidp']}(a).
• Figure 5: Four-level transition diagram with the coupling between LD and MZMs $\gamma_{1,k}$, $k=1,2,3$. Black solid lines describe the off-resonant driving fields on the transition $|e\rangle\leftrightarrow|\gamma_{1,k}\rangle$ with the detuning $\Delta_e(t)-\tilde{\Delta}_k(t)$, the Rabi-frequency $\Omega_k(t)$, and the phase $\varphi_k$. The gray dashed lines imply the population transfer among MZMs along a clockwise or counterclockwise direction.