Nonadiabatic braiding of Majorana modes

Abstract

The realization and manipulation of Majorana zero modes have drawn significant attention for their crucial role in enabling topological quantum computation. Conventional approaches to the braiding of Majorana zero modes rely on adiabatic processes. In this work, using a composite 2-Kitaev-chain system accommodating Majorana zero modes as a working example, we propose a nonadiabatic and non-Abelian geometry phase-based protocol to execute operations on these Majorana zero modes. This is possible by locally coupling the edge sites of both quantum chains with an embedded lattice defect, successfully simulating the braiding operation of two Majorana modes in a highly nonadiabatic fashion. To further enhance the robustness against control imperfections, we apply a multiple-pulse composite strategy to our quantum chain setting for second-order protection of the braiding operations. Our proposal can also support the fast and robust realization of the π/8 gate, an essential ingredient for universal quantum computation. This work hence offers a potential pathway towards the nonadiabatic and fault-tolerant control of Majorana zero modes.

Figures (3)

• Figure 1: (a) Schematic diagram of the 2-quantum-chain configuration coupled with an inserted lattice defect (LD, green circle) through the local driving. (b) Illustration of the 3-level $\Lambda$-type Rabi oscillation occuring in our nonadiabatic braiding protocol.
• Figure 2: (a) Fidelity evolution of an initial $\gamma_{1,\,R}$ mode compared with itself (red curve) and the $\gamma_{2,\, L}$ state (blue curve) within one protocol for braiding. The parameters for calculation are $N=100$, $t = \Delta = 0.1$, $\varepsilon_{d} =3.0$, $\mu/t = 0.2$, $\theta = \phi = \pi/2$ and $\Omega(t) = \frac{\pi}{T} \sin^2(\frac{2\pi t}{T})$. In our simulations, we also set $T=20$ in dimensionless units. (b) Final fidelity loss $(1-F)$ as function of the local perturbation intensity $V_s$. The local perturbation term is introduced into both chains, which has the Gaussian form of $\Delta H=\sum_{n,i=1,2}V_s\exp[(n-n_0)^2/2{\sigma_0}^2] \,c^{\dagger}_{n,i}c_{n,i}$. Here we take $n_0 = 50$ and $\sigma_0=3$ for simulation and this result holds in general for other choices of $n_0$ and $\sigma_0$.
• Figure 3: (a) Braiding fidelity evolution of an initial $\gamma_{1,\,R}$ mode with the application of composite gate method. During the period of $0\sim T$ ($2T\sim 3T$) and $T\sim 2T$ ($3T\sim 4T$), the segmented gate $U_{S}(\theta, \phi)$ ($U_{S}(\pi-\theta, \phi)$) is implemented. Here we choose the gate coefficients as $\theta=\pi/4$, $\phi=0$ and $T'=T/2$. Other parameters for calculation are the same as in Fig. 2(a). (b) Same as (a) but in the presence of random local coupling errors with control error strength $\delta_{0} = 0.06$, averaged over 100 times of implementation. The final fidelity loss is about $0.6\%$. (c) Final fidelity loss for braiding $1-F$ vs. the chemical potential $\mu$ after implementing one-site truncation. (d) $1-F$ vs. the driving error strength $\delta_0$, with (green curve) and without (orange) the usage of composite gate method. Other system parameters are the same as in Fig. 2.