Error generation and propagation in Majorana-based topological qubits

We investigate dynamical evolution of a topological memory that consists of two $p$-wave superconducting wires separated by a nontopological junction, focusing on the primary errors (i.e., qubit loss) and secondary errors (bit and phase flip) that arise due to nonadiabaticity. On the question of qubit loss we examine the system's response to both periodic boundary driving and deliberate shuttling of the Majorana bound states. In the former scenario, we show how the frequency-dependent rate of qubit loss is strongly correlated with the local density of states at the edge of wire, a fact that can make systems with a larger gap more susceptible to high-frequency noise. In the second scenario we confirm previous predictions concerning superadiabaticity and critical velocity, but see no evidence that the coordinated movement of edge boundaries reduces qubit loss. Our analysis on secondary bit-flip errors shows that it is necessary that nonadiabaticity occurs in both wires and that interwire tunneling be present for this error channel to be open. We also demonstrate how such processes can be minimized by disordering central regions of both wires. Finally, we identify an error channel for phase-flip errors, which can occur due to mismatches in the energies of states with bulk excitations. In the noninteracting system considered here, this error systematically opposes the expected phase rotation due to finite-size splitting in the qubit subspace.

Figure 1

Top: a schematic illustration of the Kitaev chain used to encode a single topological qubit. Majorana bound states (MBS) are localized at the boundaries between topological and nontopological regions. Bottom: the chain is in a topological phase in regions where $|\tilde{\mu }|<2|w|$ (the shaded area) and in a nontopological phase in regions where $|\tilde{\mu }|>2|w|$. The MBS at $x_{\text{wall}}^{\left(i\right)}$ is manipulated by moving the boundary wall with a velocity profile $v_{\text{wall}}^{\left(i\right)}\left(t\right)=d{x}_{\text{wall}}^{\left(i\right)}\left(t\right)/dt$.

Figure 2

(a) The time-averaged qubit loss, as a function of the wall velocity parameters $v_{\text{max}}$ and $\omega$. The horizontal black line shows the critical velocity $v_{\text{crit}}=\left|\mathrm{\Delta }\right|=0.3$. To generate this figure we used $L=200,a=1,m=0.5,v\left(t\right)=v_{\text{max}}sin\left(\omega t\right)$, and $\mu =0.5$. The boundary potentials in this case were set to be essentially infinite (${\mu }_{\text{boundary}}=-2000$) and the wall gradient with $\sigma =1$ (see Appendix pp2 for details on wall parametrization). (b) A cross section of (a) at $v_{\text{max}}=0.02$. We see that the maximum qubit-loss rate occurs when the frequency is close to $E_{\text{gap}}$, but where the maximum value decays as $1/E_{\text{gap}}$. The low-frequency regime corresponds to the superadiabatic limit with the rate qubit-loss error dropping off exponentially in this regime. For higher frequencies we see that error rate drops off as $1/{\omega }^4$. Crucially, because the maximum resonant frequency scales with $E_{\text{gap}}$, in the high-frequency regime, increasing the topological order parameter can make the system more susceptible to errors.

Figure 3

(a) Qubit loss for $v_{\text{max}}>v_{\text{crit}}$ (left panel) and $v_{\text{max}}<v_{\text{crit}}$ (right panel). The corresponding wall velocity profiles are plotted above. We see that the final qubit loss is larger for two moving walls than for a single moving wall (blue lines vs red lines) (plotted for $\mathrm{\Delta }=0.4$). (b) The final qubit loss at some time $t_{\text{fin}}$ after the walls have come to rest, plotted as a function of $\tau =1/\omega$ (for $v_{\text{max}}=0.1$ and $T=30$). (c) The final qubit loss in the limit of sudden wall acceleration ($\tau \to 0$). As $T$ gets large, we see a sharp distinction emerge between the $v_{\text{max}}>v_{\text{crit}}$ and $v_{\text{max}}<v_{\text{crit}}$ regimes (here, $v_{\text{crit}}=\left|\mathrm{\Delta }\right|=0.2$). [Other parameters: in all figures, $a=0.5, m=0.5, L=200, \mu =1.0$ (in the topological region).]

Figure 4

(a) The density wave packet generated by a single moving wall at $x_{\text{wall}}^{\left(1\right)}$ travels at a velocity approximately equal to $v_F$. We can see the wave tunneling through the nontopological barrier separating the topological phases (barrier height ${\mu }_{\text{barrier}}=-2.5$). Here, $v_{\text{max}}=0.3$, and $\omega =3$. (b) The bit-flip error $P_{\text{bit}}$ increases suddenly as the excitation produced by the movement of one wall hits another moving wall (legend shows which walls are in motion). The movement of $x_{\text{wall}}^{\left(1\right)}$ and $x_{\text{wall}}^{\left(4\right)}$ actually induces a bit- and phase-flip error to qubit. However, since we initalize the system in the $|\overline{0}\rangle$ state, we observe only the bit-flip channel. The time at which the error begins to increase can be estimated as approximately $t\approx |x_{\text{wall}}^{\left(i\right)}-x_{\text{wall}}^{\left(j\right)}|/v_F$ where $|x_{\text{wall}}^{\left(i\right)}-x_{\text{wall}}^{\left(j\right)}|$ is the distance between the two moving walls. [Other parameters for both figures: $m=0.5, a=0.5, \mathrm{\Delta }=0.4, L=200$, and $\mu =1.0$ (in the topological region).]

Figure 5

(a) Schematic showing how excitations originating at the outer walls can dynamically evolve through the system and cause a bit-flip error ($|\overline{0}\rangle \leftrightarrow |\overline{1}\rangle$ (b) Excitations originating at the edge of one wire can cause a phase error if they interact with opposite side of the same wire. (c) For the anomalous phase rotation, the moving boundary wall generates population transfer between ground states $|{\psi }_{\pm }\rangle$ and excited states. Energy mismatches between states in the bulk can then drive a rotation that either reinforces or opposes the natural rotation due to the ground-state splitting. (d) Schematic of how the zero-mode splitting affects ground and bulk states [see Eq. (4) for explanation of the notation]. Deviations from the natural phase rotation (blue) can occur because some excited states (red) have a phase rotation that acts in the opposite direction. The lowest-lying pairs of excited states will have a single fermion transferred from a Majorana mode (here the left edge mode) to an excited bulk mode, which means the splitting in these pairs is the same as the splitting between the Majorana modes in the odd-fermionic-parity sector, which is opposite to the splitting in the even-parity sector which contains the qubit states. Higher-energy states (black) can have the same occupation of the Majorana modes as the ground state and then do not contribute to the phase error as they are split in precisely the same way as the ground states.

Figure 6

Bit-flip error is suppressed by adding disorder in the middle of the left-hand wire. Here, the disordered region is of length $\sim 40$, and a shorter localization length $l$ corresponds to increased disorder. We use $v_{\text{max}}/v_{\text{crit}}=0.75$, the middle barrier height is ${\mu }_{\text{barrier}}=-2.5, \sigma =2$ and the data are averaged over 20 disorder realizations. [Other parameters used for the figure: $m=0.5, a=0.5, \mathrm{\Delta }=0.4, L=200$, and $\mu =1.0$ (in the topological region).]

Figure 7

Phase error $P_{\text{phase}}\left(t\right)$ as a result of moving both $x_{\text{wall}}^{\left(1\right)}$ and $x_{\text{wall}}^{\left(2\right)}$ with $v_{\text{max}}=0.1$ and $\omega =0.2$. In this simulation we patterned disorder in the central region of the left wire, giving rise to localization lengths $l$ in that region. In this case, the left wire was approximately 200 in length and region of disorder was of length 90. The disordered region prevents excitations from traveling between $x_{\text{wall}}^{\left(1\right)}$ and $x_{\text{wall}}^{\left(2\right)}$, resulting in a systematic reduction in the observed phase error. In this simulation we set $m=1/2, \mu =1, \mathrm{\Delta }=0.25$, resulting in a $v_F\approx 2$. The second wire on the right was approximately of length 160 and the wall positions $x_{\text{wall}}^{\left(3\right)}$ and $x_{\text{wall}}^{\left(4\right)}$ were kept constant. The barrier between wires was set so that no excitations could tunnel between the two wires.

Figure 8

Main figure and upper inset: qubit-loss-induced deviation from the natural phase oscillation tends to act in the opposite direction to the natural oscillation. The red dots indicate $\mu =0.03$ and the green $\mu =1.68$. Lower inset: the deviation $r\left(t\right)=\varphi \left(t\right)-\delta \left(t\right)$ (black) and the slopes of the fitted red lines give $\langle \dot{r}\left(t\right)\rangle$. To generate this figure, two wires of lengths 45 and 40 were used with an essentially infinite barrier between them. The continuum order parameter was set to $\mathrm{\Delta }=0.4$ in the left wire and $\mathrm{\Delta }=0.8$ in the right. The effective electron mass was set to $m=0.5$ and a lattice constant of $a=0.5$ was used. The velocity of of the left-hand boundary of the left wire was moved according to $v\left(t\right)=0.1sin\left(2t\right)$.

Figure 9

Bulk single-particle energies of the instantaneous Hamiltonian in the “moving frame” for $v<v_{\mathrm{crit}}$ and $v=v_{\mathrm{crit}}$, where the gap is closed.

Figure 10

The effective mass scales as (a) $M\propto 1/\mathrm{\Delta }$ and (b) $M\propto \sqrt{\mu }\propto k_F$. This is allows us to relate the topological gap and the effective wall mass as $E_{\text{gap}}\propto k_F^2/2M$.

Figure 11

The solid line represents the curve $\frac{M}{2}\gamma \left(t\right)v{\left(t\right)}^2$. The empty circles represent the value of $E\left(t\right)$ obtained through (D2). The value of $M\approx 1.69$ was obtained by fitting the two curves. The simulations were obtained using $L=100, \mathrm{\Delta }=0.4, \mu =1, a=0.5$, and $m=0.5$. The velocity profile was chosen as in (9), with $\tau =50, T=100$, and $v_{\text{max}}=0.2$.

Figure 12

The final qubit loss decreases exponentially in $\sqrt{\mu }$, which corresponds to moving up the electron band. This graph is produced with the lattice constant $a=0.5$, the total length of the wire to be $L=100$, effective electron mass $m=0.5, \mathrm{\Delta }=0.2$.

Figure 13

Analyzing final $P_{\text{loss}}$ as we vary $\mathrm{\Delta }$ keeping $\tau =8$ we can see past $\mathrm{\Delta }\sim 0.2,P_{\text{loss}}$ behaves nearly exponentially, where $\tau$ is the length of time the system is accelerating. These data are produced with $\mu =1, m=0.5, a=0.5$, a system of length $L=100$, and the lattice constant $a=0.5$.

Figure 14

Final $P_{\text{loss}}$ for different $\tau$ and the length of the time the system continues moving at $v_{\text{max}};T=0$. As we can see, there are two different behaviors of the system depending on $v_{\text{max}}/v_{\text{crit}}$ this graph is produced with $\mathrm{\Delta }=0.4, \mu =1$, with a system of length $L=100$, the lattice parameter is set to $a=0.5$, and the effective electron mass $m=0.5$. By examining Fig. 11 we can see that the different curves for $\tau$ display the same overall behavior, in contrast to $v_{\text{max}}/v_{\text{crit}}$, where either side of $v_{\text{max}}=v_{\text{crit}}$, we can see a different effect on $P_{\text{loss}}\left(t_{\text{fin}}\right)$.