Diabatic errors in Majorana braiding with bosonic bath

Majorana mode based topological qubits are potentially subject to diabatic errors that in principle can limit the utility of topological quantum computation. Using a combination of analytical and numerical methods we study the diabatic errors in Majorana-based topological Y junction that are coupled to a Bosonic bath in the Markovian approximation. We find that in the absence of a bath, the error can be made exponentially small with increasing braiding time, only when the time variation in the Hamiltonian is completely smooth. The presence of a dominantly dissipative Markovian bath is found to eliminate this exponential scaling of error to a power-law scaling as $T^{-1}$ with $T$ being the braiding time. However, the inclusion of relaxation improves this scaling slightly to go as $T^{-2}$. Thus, coupling of topological systems to Bosonic baths can lead to power law in braiding time diabatic errors that might limit the speed of topologically protected operations using Majorana modes.

Figure 1

Schematic description of the time dependence of the braiding Hamiltonian [Eq. (1)] using the protocol in Refs. [27, 28, 29]. The parameter $B_{\alpha }$ represents the coupling of Majoranas ${\gamma }_{\alpha }$ and ${\gamma }_0$. The topological degeneracy of the spectrum of the system is guaranteed by the requirement that at least one component of $\vec{B}=(B_x,B_y,B_z)$ vanishes. Braiding is accomplished by successive rotations about the vanishing direction chosen to be $x,y$, and $z$. The Bosonic bath considered here is assumed to couple as fluctuations of $B_{x,y,z}$.

Figure 2

Plot of different components of Majorana coupling field $\left(B_i;i\in \{x,y,z\}\right)$ appearing in Eq. (1) as a function of scaled time for $s=t/T$ as described by Eq. (10).

Figure 3

Numerical plot of log of norm of difference between time evolved and instantaneous Bloch zero vector at scaled time $s=1$ as a function of $\sqrt{T}$ for strength $s_x=s_y=0$ obtained using Eq. (27). The system is initialized in the ground state of $H_{2\mathrm{Level}}$ or equivalently the initial value of $R(s=0)$ in the Bloch equation is set as $R_0\left(0\right)=-\vec{B}\left(0\right)$. The plot shows exponential decay in $\mathcal{E}$ as a function of total time $T$ when system-bath coupling is zero consistent with the result derived in Appendix pp1. Specifically, $\mathcal{E}=\parallel R\left(1\right)-R_0\left(1\right)\parallel \sim e^{-\sqrt{T}}$.

Figure 4

Plot of $\mathcal{E}$ as a function of time $T$ for different values of effective dephasing strength $\alpha$ characterized by values of $s_x=s_y$ and absence of effective relaxation, $\beta =0$ [see Eq. (25)]. Relaxation and dephasing parameters, $\eta$ and ${\eta }_0$, that solely depend on bath Hamiltonian are set to $\eta =0$ and ${\eta }_0=1$, respectively. The solid line is obtained by solving the Bloch equation [Eq. (31)] and is compared against (dashed curve) theoretical bound given by Eqs. (42) and (41), i.e., the dashed curve is the calculated value of $log\left[(1/T)f_0\left(1\right)\right]$ using the expression given in Eq. (41). The low-T region characterized by the oscillatory section of the curves is independent of system-bath coupling strength and follows closely the numerical values obtained by solving the Bloch equation [Eq. (27)] in the absence of bath denoted by the dashed-brown-colored curve.

Figure 5

(Top left) Plot of $\mathcal{E}$ as a function of total time taken by the diabatic drive $T$ for different values of effective dephasing $\alpha$ and relaxation strength $\beta$ characterized by the shown values of $s_x=s_y$ [see Eqs. (27) and (26)]. Relaxation and dephasing parameters, $\eta$ and ${\eta }_0$, that solely depend on bath Hamiltonian are both set to $\eta ={\eta }_0=1$ for all panels in this figure. For each set of parameters, the analytical curve (dashed line) is obtained by plotting the expression in Eq. (51) with $f_0$ given by Eq. (54). The numerical curve (solid line) is obtained by numerically solving the Bloch equation in Eq. (27). The slope $m_N$ (given in legend) is calculated using the data in the neighborhood of log(T) = 29.5 on the $X$ axis for each curve. The asymptotic dependence on $T$ is given by $e^4{T}^{-2}$ [Eq. (53)], represented in the figure by the solid (thick) brown curve. (Top right) Magnified view of the top left panel in the small-T region. The values obtained for the special case of zero dephasing and relaxation is plotted using a (thick) brown dashed curve. For each coupling strength and sufficiently small values of T, the $\mathcal{E}=log(R\left(1\right)-R_0\left(1\right))$ behavior is independent of system bath coupling strength and is well captured by the exponential dependence on T (see Fig. 3). The abrupt change from the exponential to a polynomial dependence on T occurs when the analytic estimate of $\mathcal{E}$ coincides with its value under zero system-bath coupling assumption. (Bottom panels) Similar to the top panels, log norm of difference between time evolved and instantaneous Bloch vector at scaled time $s=1\left(t/Tmod3=1\right)$ as a function of log of total time $T$. However, unlike the top panels $s_x=2s_y$ and $s_y=2s_x$ for bottom left and right panels, respectively. The qualitative behavior is similar to ones observed in the top panels.

Figure 6

Plot of effective dephasing $\left(\alpha \right)$ and relaxation $\left(\beta \right)$ as a function of scaled time $s$ for system-bath coupling strength $s_x=0.5$ and $s_y$ given in the key. The dashed curve corresponds to effective relaxation $\beta$ and the solid curve corresponds to effective dephasing $\alpha$. The dephasing parameter ${\eta }_0$ and the relaxation parameter $\eta$ are chosen equal to each other and set to 1.

Figure 7

(Panel a) Plot of $\rho$ [Eq. (B12)] for different values of time $T$. The peak height and the width of the probability density $\rho$ is an increasing and a decreasing function of $T$, respectively. The probability densities look identical at this scale for values of $s_x=s_y$ varied from 0.005 to 0.5 (i.e., over two orders of magnitude). (Panels b and c) Comparison of ${〈{\int }_s^1\beta 〉}_{\rho }$ [defined by Eq. (B13)], plotted in solid curves, versus its approximate estimate given by values of ${\int }_{s_{\max }}^1\beta$ as a function of total time $T$ (in the units of $1/\mathrm{\Delta })$, plotted in dashed curves, for different values of dephasing $\alpha$ and relaxation strength $\beta$ characterized by the shown values of $s_x=s_y$ [see Eqs. (27) and (26)]. Relaxation and dephasing parameters, $\eta$ and ${\eta }_0$, that solely depend on bath Hamiltonian are both set to $\eta ={\eta }_0=1$. Both sets of curves, solid and dashed, are computed numerically. There exists a region, for small values of $T$, over which $T$ dependence of ${〈{\int }_s^1\beta 〉}_{\rho }$ vanishes (panel c). This region is well captured by the approximate formula ${\int }_{s_{\max }}^1\beta$, however the value of ${〈{\int }_s^1\beta 〉}_{\rho }$ itself is underestimated by the formula. For large values of $T,{〈{\int }_s^1\beta 〉}_{\rho }$ values tend to oscillate, however, the average slope is again well captured by the approximate formula ${\int }_{s_{\max }}^1\beta$ (top left panel). (Panels d and e) Plot of $1-s_{\max }$ as a function of $logT$ (bottom right panel) and the same plot in log-log scale is shown in the bottom left panel. Notice that $T$ independent region of ${〈{\int }_s^1\beta 〉}_{\rho }$ corresponds to $s_{\max }\approx 0.5$. $s_{\max }\approx 0.5$ region ends in a kink beyond which $1-s_{\max }$ decreases zero, asymptotically approaching $\frac{1}{logT}$. This asymptotic dependence is shown in the log-log plot in the bottom right panel.