Braiding quantum circuit based on the $4\pi$ Josephson effect

We propose a topological qubit in which braiding and readout are mediated by the $4\pi$ Majorana-Josephson effect. The braidonium device consists of three Majorana nanowires that come together to make a trijunction. In order to control the superconducting phase differences at the trijunction, the nanowires are enclosed in a ring made of a conventional superconductor. In order to perform initialization and readout, one of the nanowires is coupled to a fluxonium qubit through a topological Josephson junction. We analyze how flux-based control and readout protocols can be used to demonstrate braiding and qubit operation for realistic materials and circuit parameters.

Figure 1

(a) Topological Josephson trijunction ring. The light blue segments are $p$-wave wires. The MBSs are marked by blue dots and labeled ${\gamma }_n$. The phase difference between superconductors $i$ and $j$ is labeled ${\varphi }_{ij}$, while the external flux through the loop formed by superconductors $i$ and $j$ is labeled ${\mathrm{\Phi }}_{ij}$. The dashed yellow lines show the electron basis used in the text labeled $c_n$. (b) Braiding sequence with the location of the MBSs (red and green dots) and the phase differences ${\varphi }_{ij}$. The inner MBS coupling is depicted by an orange bar. (c) The solid red line shows the braiding path that is required by the flux corral. The Hilbert space trajectory along the solid path is equivalent to going twice around the dashed red triangle. (d) The probability of being in a specific state $x^{\pm }\in \{a,b,c,d\}$ during the double-braid process.

Figure 2

Probability of being in the expected state $|a^{-}\rangle$ after the double braid plotted against the log of the duration $\tau$ of each step (in units of $1/\alpha$) for different values of $\alpha$. The horizontal dashed black line marks perfect braiding. The vertical dashed lines mark the timescales $\tau =1/\alpha$ (black) and $\tau =E_M/6{\alpha }^2$ (color corresponding to solid curve).

Figure 3

Full braidonium device. A fluxonium loop is coupled to the transmission line via a microwave resonator. The fluxonium loop is coupled to the trijunction loop through the $p$-wave wire (light blue lines). External flux (fast) through the fluxonium loop and each of the three sections of the braiding ring is controlled by on-chip coils. Several electrostatic gates (slow) are used to tune the superconducting nanowires into the topological regime. The locations of MBSs are marked by blue dots. Uncovered regions of semiconductor nanowire used for the Josephson junctions are depicted as green lines.

Figure 4

(a) Energy levels as a function of ${\mathrm{\Phi }}_{24}$. The color of the curve reflects its electron occupation. Each curve represents two nearly degenerate fermion occupations. Red is both $|a^{+}\rangle$ and $|d^{+}\rangle$, blue is $|a^{-}\rangle$ and $|d^{-}\rangle$, orange is $|b^{+}\rangle$ and $|c^{+}\rangle$, and purple is $|b^{-}\rangle$ and $|c^{-}\rangle$. The dashed lines have the same color scheme as the solid lines, but the fluxonium is in the first excited state. (b) Energy difference between states $|0,x^{-}\rangle$ and $|1,x^{-}\rangle$ (blue) and states $|0,x^{+}\rangle$ and $|1,x^{+}\rangle$ (red).

Figure 5

Probability of being in a specific state during the double-braid process for different initial states. For (a), we start in the state $1/\sqrt{2}(|a^{+}\rangle +|d^{+}\rangle )$, while in (b)–(d) we start in eigenstates $|b^{+}\rangle , |c^{+}\rangle$, and $|d^{+}\rangle$, respectively. A similar plot in which the initial state is $|a^{+}\rangle$ is shown in Fig. 1.

Figure 6

Depiction of the trijunction during the first step of the braiding procedure. The purple numbers show the desired phase difference between nanowires before and after the step. The goal is to move the Majorana at ${\gamma }_{g2}$ to ${\gamma }_{g3}$. Flux error could instead cause the Majorana at ${\gamma }_{g1}$ to move to ${\gamma }_{g3}$.

Figure 7

(a) A path through the space of phase differences that does not go straight from one point to the next. The parameter $\delta$ characterizes the point along the curve that is farthest from the straight line. (b) The fidelity of the path depicted in (a) as a function of the $\delta$ parameter. The fidelity is flat from about $-\pi /16$ to $\pi /16$, as shown in the inset. (c) A path through the space of phase differences that does not hit the vertices at the end of each step. Here, ${\delta }_i$ measures the distance between the point that is hit by the path at the end of step $i$ and the target point. These parameters ${\delta }_i$ are generated from a random Gaussian distribution with a standard deviation of $\sigma$. (d) The fidelity, averaged over 100 trials, of the path depicted in (c) as a function of the standard deviation $\sigma$.

Figure 8

Controlling the phase difference with the external flux. (a) and (c) show the energy levels of the braiding ring as a function of the external flux ${\mathrm{\Phi }}_{12}=\pi -{\mathrm{\Phi }}_{23}$, while ${\mathrm{\Phi }}_{31}=-\pi$ for (a) $E_J^t=3.0E_M, E_C^t=0.1E_M$, and $E_L^t=1.0E_M$ and (c) $E_J^t=3.0E_M, E_C^t=0.1E_M$, and $E_L^t=5.0E_M$. (b) and (d) show the probability distribution of the ground state as a function of the phase differences ${\varphi }_{12}$ and ${\varphi }_{23}$, corresponding to the red arrows in (a) and (c), respectively. (e) shows the $F$ factor, which is a measure of the probability of being in the correct state (${\varphi }_{12}={\mathrm{\Phi }}_{12}$ and ${\varphi }_{23}={\mathrm{\Phi }}_{23}$) as a function of the inductive energy.