Symmetry-protected non-Abelian braiding of Majorana Kramers pairs

We develop a complete theory for symmetry protected non-Abelian statistics of Majorana Kramers' pairs (MKPs) in time-reversal (TR) invariant topological superconductors, with fundamental results being uncovered. By introducing an effective Hamiltonian approach to describe the braiding of MKPs, we show that the non-Abelian braiding is protected when the effective Hamiltonian exhibits a new TR-like antiunitary symmetry, which is satisfied if the system is free of dynamical noise. Importantly, even the dynamical noise cannot cause error in braiding, unless the noise correlation function breaks a dynamical TR symmetry. This is a profound result and generalizes the TR symmetry protection of MKPs to the dynamical regime. Moreover, the resulted error by noise is shown to be a higher-order effect, compared with the decoherence of Majorana qubits without TR symmetry protection. This study completes the theory of symmetry-protected non-Abelian statistics of MKPs, and shows that the non-Abelian braiding of MKPs is well observable and may have versatile applications to future quantum computation technologies.

Figure 1

Non-Abelian braiding of MKPs immune to static disorder scattering. (a) Sketch of the braiding process, with 1,2,3 denoting the sequence of transporting MKPs by gate control. (b)–(d) Evolution of MKP wave function in a full braiding with different disorder strengths $W_0$ (in unit of bulk gap $E_g)$. The non-Abelian braiding of MKPs is confirmed by that the magnitude $\eta \left(t\right)=\langle {\gamma }_1\left(0\right)|{\gamma }_1\left(t\right)\rangle {|}_{t=2T}=-1$ after a full braiding at $t=2T$ (similar for other Majorana modes) [7, 38, 39]. The adiabatic condition is satisfied in that $\zeta \left(t\right)={\sum }_{j=1,2}\left[\right|\langle {\gamma }_1\left(t\right)|{\gamma }_j\left(0\right)\rangle {|}^2+|\langle {\gamma }_1\left(t\right)|{\tilde{\gamma }}_j\left(0\right)\rangle {|}^2]$ returns to unity after a single $\left(t=T\right)$ and full $\left(t=2T\right)$ braiding.

Figure 2

Numerical simulation of error in braiding $\mathcal{D}\left(T\right)$ induced by uniform dynamical noise with strength $V_0$. The bulk gap $E_g=0.58$. Note that for $\delta \phi =0$ and $\pi$, the local rotations and braiding error disappear.

Figure 3

(a) Error in braiding $\mathcal{D}\left(T\right)$ induced by random dynamical noise vs disorder strength $V_0$ and coherence length $l_0$. (b) The fidelity of the braiding $\mathcal{F}\left(T\right)=|\langle {\gamma }_a|e^{-i{H}_{E}T}|{\gamma }_a{\rangle |}^2$. The gap $E_g\approx 0.58$ and Majorana localization length ${\lambda }_M\sim 20$ sites in the numerical simulation.

Figure 4

(a)–(d) Braiding of two MKPs for a single segment of a topological superconductor. The sketched transporting paths in (e) and (f) govern the effective local rotations of the MKPs ${\gamma }_2,{\tilde{\gamma }}_2$ and ${\gamma }_1,{\tilde{\gamma }}_1$, respectively, during the braiding. It can be seen that the two paths are inverse to each other and thus give rise to the opposite local rotations for the two MKPs if the system has no dynamical noise.

Figure 5

(a)–(d) Braiding of two MKPs on two separated segments of a topological superconductor. The sketched transporting paths in (e) and (f) govern the effective local rotations of the MKPs ${\gamma }_2,{\tilde{\gamma }}_2$ and ${\gamma }_1,{\tilde{\gamma }}_1$, respectively, during the braiding. Similar to the cases in Fig. 4, the two paths are inverse to each other and thus give rise to the opposite local rotations for the two MKPs.

Figure 6

Non-Abelian braiding of MKPs immune to static disorder scattering. The setup sketched in Fig. 5 (1-d) is used for the numerical simulation. (a)–(i) The evolution of MKP wave function in a full braiding with different disorder strengths $W_0$ $(E_g$ is the bulk gap of the TSC). The non-Abelian braiding of MKPs is confirmed by that the magnitude $\eta (t=2T)=\langle {\gamma }_1\left(0\right)|{\gamma }_1\left(t\right)\rangle {|}_{t=2T}=-1$ after a full braiding at $t=2T$ (similar for other Majorana modes). The adiabatic condition is satisfied in that the projection $\zeta \left(t\right)={\sum }_{j=1,2}\left[\right|\langle {\gamma }_1\left(t\right)|{\gamma }_j\left(0\right)\rangle {|}^2+|\langle {\gamma }_1\left(t\right)|{\tilde{\gamma }}_j\left(0\right)\rangle {|}^2]$ returns to unity after a single $\left(t=T\right)$ and full $\left(t=2T\right)$ braiding.