Majorana qubit readout using longitudinal qubit-resonator interaction

We propose a quantum nondemolition Majorana qubit readout protocol based on parametric modulation of a longitudinal interaction between a pair of Majorana bound states and a resonator. The interaction can be modulated through microwave-frequency gate voltage or flux control of a tunable tunnel barrier. The qubit-resonator coupling is quantum nondemolition to exponential accuracy, a property inherited from the topological nature of the Majorana zero modes. The same mechanism as used for single-qubit readout can also be extended to perform multiqubit measurements and be used to enact long-range entangling gates.

Figure 1

Schematic Majorana qubit setup. Two topological superconducting wires (TS) host four Majorna bound states, located at the respective wire ends (crosses). A dot is formed in a nonproximitized semiconducting segment (Sm), acting as a tunable barrier between the two wires. A conventional superconductor (S) shunts the two wires such that they behave as a single superconducting island with a uniform superconducting phase. The superconducting island furthermore couples capacitively to the voltage of a resonator (${\widehat{V}}_r$). The overlap of the Majorana wave functions can be controlled through gate voltages ($V_g$) or through an external flux threading the superconducting loop (${\mathrm{\Phi }}_x$). A geometry with two parallel wires as in Ref. [8] can also be used.

Figure 2

Longitudinal readout concept: A semiconducting segment (Sm) forms a dot acting as a tunable barrier between the topological quantum wires. By modulating either the tunneling amplitudes, barrier excitation energy, or an external flux through the qubit loop, the longitudinal coupling can be modulated with an amplitude ${\tilde{g}}_z$, leading to a qubit-state dependent displacement of the resonator with steady-state magnitude $\pm {\tilde{g}}_z/\kappa$.

Figure 3

(a)–(c) Parametric dependence of longitudinal coupling on $\delta /t \left(a\right)$ and $t/\delta \left(b\right)$ for ${\phi }_x=0$, and on ${\phi }_x \left(c\right)$ for $t/\delta =1$ (solid line) and $t/\delta =0.5$ (dotted line). (d), (e) Qubit splitting $\hslash {\omega }_q/t \left(d\right), \hslash {\omega }_q/\delta (e,f)$ for the same parameters as in the top row. The dashed lines in (a), (b), (d), and (e) are approximations for small $t/\delta$. The parameter ${\delta }_{\lambda }={\lambda }_0-{\lambda }_C$ is the difference in resonator-coupling strength for the semiconducting barrier and the superconducting island.

Figure 4

Measurement infidelity for ideal longitudinal readout with interaction ${\widehat{H}}_{\text{ideal}}$. $\left(a\right)$ Infidelity as a function of modulation amplitude for two different measurement times $\kappa \tau =1,2$. $\left(b\right)$ Measurement time needed to reach infidelities $1-F={10}^{-3}$ and $1-F={10}^{-6}$ as a function of modulation amplitude.

Figure 5

Schematic of two Majorana box qubits interfaced by two separate semiconducting barriers ($b_1,b_2$), and capacitively coupled to a common readout resonator voltage ${\widehat{V}}_r$. In the figure, capacitive coupling to each superconducting island is indicated, but it is sufficient that one of the two qubits couple to the resonator.