Constructing a universal operation scheme for Majorana qubits remains a central issue for the topological quantum computation. We study the Landau-Zener-Stückelberg interference in a Majorana qubit and show that this interference can be used to achieve controllable operations. The Majorana qubit consists of an rf SQUID with a topological nanowire Josephson junction which hosts Majorana bound states. In the SQUID, a magnetic flux pulse can drive the quantum evolution of the Majorana qubit. The qubit experiences two Landau-Zener transitions when the amplitude of the pulse is tuned around the superconducting flux quanta 2e/ . The Landau-Zener-Stückelberg interference between the two transitions rotates the Majorana qubit, with the angle controlled by the time scale of the pulse. This rotation operation implements a high-speed single-qubit gate on the Majorana qubit, which is a necessary ingredient for the topological quantum computation.

Stimulated by a recent experiment observing successfully two superconducting states with even- and odd-number of electrons in a nanowire topological superconductor as expected from the existence of two end Majorana quasiparticles (MQs) [Albrecht et al., Nature 531, 206 (2016)], we propose a way to manipulate Majorana qubit exploiting quantum tunneling effects. The prototype setup consists of two one-dimensional (1D) topological superconductors coupled by a tunneling junction which can be controlled by gate voltage. We show that the time evolution of superconducting phase difference at the junction under a voltage bias induces an oscillation in energy levels of the Majorana parity states, whereas the level-crossing is avoided by a small coupling energy of MQs in the individual 1D superconductors. This results in a Landau-Zener-Stückelberg (LZS) interference between the Majorana parity states. Adjusting pulses of bias voltage and gate voltage, one can construct a LZS interferometry which provides an arbitrary manipulation of the Majorana qubit.

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We propose a superconducting flux qubit in a dc SQUID structure, formed by a conventional insulator Josephson junction and a topological nanowire Josephson junction with Majorana bound states. The zero-energy Majorana bound states transport 4π period Josephson currents in the nanowire junction. The interplay between this 4π period Josephson effect and the convectional 2π period Josephson effect in the insulator junction induces a double-well potential energy landscape in the SQUID. As a result, the two lowest-energy levels of the SQUID are isolated from the other levels. These two levels show counterpropagating supercurrents, thus can be used as a flux qubit. We reveal that this flux qubit has the merits of stability to external noises, tolerance to the deviation of system parameters, and scalability to large numbers. Furthermore, we demonstrate how to couple this flux qubit with the Majorana qubit by tuning the junction parameters, and how to use this coupling to manipulate the Majorana qubit.

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