Braiding errors in interacting Majorana quantum wires

Avenues of Majorana bound states (MBSs) have become one of the primary directions towards a possible realization of topological quantum computation. For a Y junction of Kitaev quantum wires, we numerically investigate the braiding of MBSs while considering the full quasiparticle background. The two central sources of braiding errors are found to be the fidelity loss due to the incomplete adiabaticity of the braiding operation as well as the finite hybridization of the MBSs. The explicit extraction of the braiding phase from the full many-particle states allows us to analyze the breakdown of the independent-particle picture of Majorana braiding. Furthermore, we find nearest-neighbor interactions to significantly affect the braiding performance for better or worse, depending on the sign and magnitude of the coupling.

Figure 1

Y junction of three Kitaev chains and the braiding protocol considered here. MBSs, depicted by red and yellow spheres, reside at the ends of segments with a topological nontrivial phase (highlighted by blue tubes). (a) The initial state. Geometric angles ${\phi }_i$ also correspond to the superconducting pairing phases of the respective junction legs. (b) The exchange “braiding” protocol (left to right and top to bottom), as in Ref. [26]. Red (green) arrows indicate displacements of MBSs by contraction (extension) of the topological chain segment, implemented by locally ramping up (down) the local potential over the critical value ${\mu }_c=2J$. At the end of the braiding protocol (right picture, bottom row), the same chain segments as in the initial state reside in the topological phase, but the MBSs have been exchanged.

Figure 2

Time evolution of the acquired exchange phases ${\varphi }_{\mathrm{g}}^{e/o}(t,\phi )$ (dashed/dotted lines), and the phase difference $\mathrm{\Phi }(t,\phi )$ (solid lines) during the braiding protocol for sine-squared (top) and guillotine ramps (bottom). We consider a Y junction with equal-size legs $\ell =5$, for triplet-pairing phases $\phi =\pi /6,\pi /3,\pi /2$ and amplitude $\mathrm{\Delta }=1$. The braiding protocol takes a total time $t_{\mathrm{f}}=6T$, where we choose a time period $T=750$ for each ramp-up/down step, with $\alpha =0.025$. After complete execution of the braiding exchange, upon reaching $t=t_{\mathrm{f}}$ (dashed-dotted vertical line), an extra time evolution of duration $T$ is carried out with fixed Hamiltonian $H\left(t_{\mathrm{f}}\right)$. The insets show the residual dynamic evolution of the exchange phase $\mathrm{\Phi }\left(t\right)$, for $t\in [5.8T,7T]$, zoomed in on the $y$ scale.

Figure 3

Localization length $\xi$ and hybridization energy $\varepsilon$ of Majorana edge modes in a Kitaev chain with $\mu =0$. (a) MBS localization length for infinite-size, $\xi \left(\mathrm{\Delta }\right)$, and finite-size Kitaev chains, $\xi (\mathrm{\Delta },L)$, as a function of pairing amplitude $\mathrm{\Delta }$. Results for the infinitely long (bold gray curve) and finite Kitaev chains match closely already for modest $L=11$ (dashed curve) and $L=6$ (dashed-dotted curve). (b) MBS hybridization energy $\varepsilon (\mathrm{\Delta },L)$ for finite-size Kitaev chains. (c) Semilog plot of scaled MBS hybridization energy $\varepsilon (\mathrm{\Delta },L)/4\mathrm{\Delta }$ vs scaled inverse localization length $L/\xi (\mathrm{\Delta },L)$. For comparable $L$ and $\xi (\mathrm{\Delta },L)$ there is a clear deviation from the exponential dependence (upper left corner).

Figure 4

Exchange phase $\mathrm{\Phi }(\xi ;\mathrm{\Delta },\phi ,T)$ (top row) and loss of fidelity $w_{\mathrm{loss}}^{e/o}(\xi ;\mathrm{\Delta },\phi ,T)$ (bottom row) vs MBS localization length $\xi \left(\mathrm{\Delta }\right)$, for sine-squared (black diamonds) and guillotine ramps (red circles). We consider a symmetric Y junction $\left(\phi =\pi /3\right)$ with equal-size legs $\ell =5$. Plots from left to right correspond to execution of the braiding protocol with ramping time periods of $T=250$, 500, 750, 1500, and 3000, respectively, with $\alpha =0.025$. Data points (left to right) for each plot correspond to pairing amplitudes $\mathrm{\Delta }=1.0,\cdots ,0.3$ in steps 0.05, which translates to increasing MBS localization length $\xi \left(\mathrm{\Delta }\right)$ from left to right. The loss of fidelity (bottom row) in even and odd total-parity sectors is shown by solid and open symbols, respectively.

Figure 5

Exchange phase $\mathrm{\Phi }(\xi ;\mathrm{\Delta },\phi ,T)$ vs MBS localization length $\xi \left(\mathrm{\Delta }\right)$, for unitary (black diamonds), ground-state projected (cyan crosses), and adiabatic braiding process (blue pluses). As in Fig. 4, the braiding protocol is applied to a symmetric Y junction $\left(\phi =\pi /3\right)$ with equal-size legs $\ell =5$, using the sine-squared ramp with time period $T=750$ and $\alpha =0.025$. Data points (left to right) for each plot correspond to pairing amplitudes $\mathrm{\Delta }=1.0,\cdots ,0.3$ in steps of 0.05, which translates to increasing MBS localization length $\xi \left(\mathrm{\Delta }\right)$ from left to right. Lines are guides to the eye.

Figure 6

Exchange phase $\mathrm{\Phi }(V;\mathrm{\Delta },\phi ,T)$ (upper plot) and loss of fidelity $w_{\mathrm{loss}}^{e/o}(V;\mathrm{\Delta },\phi ,T)$ (middle plot) vs interaction strength $V$ for pairing amplitudes $\mathrm{\Delta }=1.0$ (black diamonds), $\mathrm{\Delta }=0.8$ (red circles), and $\mathrm{\Delta }=1.1$ (cyan squares). As in Fig. 4, the braiding protocol is applied to a symmetric Y junction $\left(\phi =\pi /3\right)$ with equal-size legs $\ell =5$, using the sine-squared ramp with time period $T=750$ and $\alpha =0.025$. The bottom plot shows the MBS localization length $\xi (V;\mathrm{\Delta },L)$ vs interaction strength $V$ for a Kitaev chain with $L=22$, 11, and 6 sites (solid, dashed, and dotted lines). Again, we consider pairing amplitudes $\mathrm{\Delta }=1.0$ (black), $\mathrm{\Delta }=0.8$ (red), and $\mathrm{\Delta }=1.1$ (cyan).