Braiding-based quantum control of a Majorana qubit built from quantum dots

Topology-related ideas might lead to noise-resilient quantum computing. For example, it is expected that the slow spatial exchange (“braiding”) of Majorana zero modes in superconductors yields quantum gates that are robust against disorder. Here, we report our numerical experiments, which describe the dynamics of a Majorana qubit built from quantum dots controlled by time-dependent gate voltages. Our protocol incorporates nonprotected control, braiding-based protected control, and readout of the Majorana qubit. We use the Kitaev chain model for the simulations, and we focus on the case when the main source of errors is quasistatic charge noise affecting the hybridization energy splitting of the Majorana modes. We provide quantitative guidelines to suppress both diabatic errors and disorder-induced qubit dephasing, such that a fidelity plateau is observed as the hallmark of the topological quantum gate. Our simulations predict realistic features that are expected to be seen in future braiding experiments with Majorana zero modes and other topological qubit architectures.

Figure 1

Protocol to demonstrate a braiding-based $\pi /2$ gate on a Majorana qubit. (a) Blue: $Y$ junction built from Kitaev chains. Red: a straight Kitaev wire. Blue and red together form a Majorana qubit. Black: readout dot. (b) The five-step protocol, analogous to a Ramsey protocol, aiming to detect the braiding-based $\pi /2$ gate: A, Initialization; B, First $\pi /2$ pulse; C, Braiding; D, Second $\pi /2$ pulse; E, Readout. (c) Procedure of a single MZM exchange used in step C, Braiding.

Figure 2

Ramsey-type experiment to detect the braiding-based $\pi /2$ gate on the Majorana qubit. Error-free evolution of the qubit Bloch vector, shown after each step (A)–(D) of the protocol, cf. Table 1. Qubit basis states are $|\text{e,e}\rangle$ and $|\text{o,o}\rangle$. The last column shows the expected value of the readout dot occupation at the and of the protocol. The three paths correspond to the protocol with 0, 1, and 2 exchanges.

Figure 3

In a disorder-free system, a braiding-based $\pi /2$ gate implies a fidelity plateau for long braiding times. Readout dot occupation is shown as a function of the braiding time, for the setup with eight quantum dots ($N_{\text{c}}=1$), obtained from our numerical simulation of a disorder-free system. (a) Protocol with one exchange of the Majorana zero modes in the $Y$ junction. (b) Protocol with two exchanges. For short braiding times, braiding does not affect the dynamics resulting in $n_{\text{R}}\approx 1$. For intermediate braiding times, diabatic errors dominate the data. In the adiabatic limit ($T_{\text{B}}\to \infty$), readout dot occupation shows a fidelity plateau at (a) 1/2 and (b) 0, as predicted in Table 1.

Figure 4

Diabatic errors of the braiding-based $\pi /2$ gate scale with the ramping time. Diabatic error is illustrated as a function of the ramping time in the case of the $2\times$ exchange protocol, for three different chain lengths. For this protocol, diabatic error corresponds to the readout dot occupation at the end of the protocol. The three curves overlap, showing that the diabatic error scales with the ramping time (which is inversely proportional to the domain-wall speed), irrespective of the chain length. For longer ramping times $T\gtrsim 100\hslash /v$, the $n_{\text{R}}$ curves saturate due to the nonzero error of tunnel-pulse-based gates and readout.

Figure 5

Numerical simulation of the braiding experiment in the presence of on-site disorder. Disorder-averaged final-state occupation ${\overline{n}}_{\text{R}}$ of the readout dot is shown as a function of the braiding time, for the smallest system size $N_c=1$ with disorder strength ${\sigma }_{\mu }=0.04$. Each curve is a result of $N_{\text{r}}=1000$ realizations, dots show the mean of the occupations, while error bars depict the 10–$90\%$ quantiles. (a) Single exchange. Disorder-averaged readout dot occupation for long braiding time is similar to that without disorder [see Fig. 3], but the error bars arise due to disorder. (b) Double exchange. The disorder changes the mean of the readout dot occupation in the adiabatic limit [cf. Fig. 3], and the error bars also show the influence of disorder. The plateau between 100 and 1000 can serve as the signature of successful braiding in a real experiment.

Figure 6

Demonstration of topological protection. Disorder-averaged occupation of the readout dot as a function of the braiding time is shown in the double-exchange protocol (a) for a given disorder strength ${\sigma }_{\mu }=0.15v$ and for different system sizes; (b) for the smallest system size $N_{\text{c}}=1$ and for different disorder strengths. Each curve is obtained by averaging for $N_{\text{r}}=1000$ realizations. For (a) increasing system size, and (b) for decreasing disorder strength, a plateau with ${\overline{n}}_{\text{R}}\approx 0$ appears for long braiding times, and the plateau length increases.