Majorana braiding dynamics in nanowires

Superconductors hosting long-sought excitations called Majorana fermions may ultimately be used as qubits of fault-tolerant topological quantum computers. A crucial challenge for the development of practical topological quantum computers is the implementation of quantum operations of nearly degenerate quantum states as a dynamical process of Majorana fermions. In this paper, we investigate the braiding dynamics of Majorana fermions on superconducting nanowires. In a finite-size system, a nonadiabatic dynamical process between nearly degenerate states dominates the braiding that operates qubits of Majorana fermions. Our simulations clarify how qubits behave in the real-time braiding process and elucidate the optimum superconducting nanowire conditions for efficient topological quantum operation.

Figure 1

(a) Cross-shaped topological superconducting nanowire. The orange junctions connecting to the central site are cut or linked through gate potentials, effectively leaving four independently controlled wires. When gates 2 and 4 are connected, while 1 and 3 are shut, six Majorana end modes ${\gamma }_i(i=1,2,3,4,5,6)$ are obtained. (b) A typical energy spectrum of Majorana bound states as a function of $t$, where the time dependence is given by the gating process in Fig. 2. Here we take ${\mathrm{\Delta }}_{\mathrm{s}}=\lambda , \mu =0.7\lambda$, and $N=20$. The color of the lines match the color of the modes in Fig. 1. The finite coupling of Majorana bound state on the same wire slightly lifts the zero-mode degeneracy.

Figure 2

(a)–(h) Braiding process of Majorana fermions. The upper half of each panel shows how the gates are manipulated and the bottom half gives the simulation results for ${\gamma }_1$ and ${\gamma }_2$ with $T=100/{\mathrm{\Delta }}_{\mathrm{s}}, {\mathrm{\Delta }}_{\mathrm{s}}=\lambda , \mu =0.7\lambda ,$ and $N=20$. (i) The projection of the wave function at each instant on the two eigenstates defined by ${\gamma }_1$ and ${\gamma }_2$ shows the transition from one to another upon braiding. (j) The gate manipulation depicted in panels (a)–(h).

Figure 3

Success rate $P_{\mathrm{s}}$ of the Majorana braiding. Each line accounts for a different length (in number of sites) of each wire (1–4), with the abscissa being our operation time $T$ and the ordinate being the success rate $P_s$. We take ${\mathrm{\Delta }}_{\mathrm{s}}=0.1\lambda$ and $\mu =0.7\lambda$. The inset shows $P_{\mathrm{s}}$ for ${\mathrm{\Delta }}_{\mathrm{s}}=\lambda$. The red lines (80 and 20 sites/wire) denote a clear separation between its adiabatic (slow, large-$T$) and dissipative (fast, small-$T$) domains, suggesting an appreciable necessity for both size- and time-scale adjustment. $\epsilon$ indicates the order of the coupling energy retrieved numerically.

Figure 4

Comparison of the projection of the wave function on the eigenstates during the braiding process for ${\mathrm{\Delta }}_{\mathrm{s}}=\lambda , \mu =0.7\lambda$, and $N=20$. Panels (a), (b), and (c) respectively show the projections on the eigenstates with Majorana modes ${\gamma }_5$ and ${\gamma }_6, {\gamma }_3$ and ${\gamma }_4$, and ${\gamma }_1$ and ${\gamma }_2$. ${\gamma }_1$ corresponds to the mode being braided.

Figure 5

Behavior outside (good) braiding conditions. (a) and (b) represent the evolution of the system at the adiabatic limit ($T=100000/{\mathrm{\Delta }}_{\mathrm{s}}$), where both Majorana edge states migrate to the next wire, consequently remaining on the same state, as can be observed from the projection of the wave function on the instantaneous eigenstates in (c), where only $E_3$ line stretches at 1. (d) and (e) account for the opposite, fast regime ($T=1/{\mathrm{\Delta }}_{\mathrm{s}}$), with bulk excitations visible and a more erratic spread over the eigenstates in (f). Line correspondence is the same in (f) and (c). We take ${\mathrm{\Delta }}_{\mathrm{s}}=\lambda , \mu =0.7\lambda$, and $N=20$.