Majorana bound states have been a focus of condensed matter research for their potential applications in topological quantum computation. Here we utilize two charge-qubit arrays to explicitly simulate a DIII class one-dimensional superconductor model where Majorana end states can appear. Combined with one braiding operation, universal single-qubit operations on a Majorana-based qubit can be implemented by a controllable inductive coupling between two charge qubits at the ends of the arrays. We further show that in a similar way, a controlled-not gate for two topological qubits can be simulated in four charge-qubit arrays. Although the current scheme may not truly realize topological quantum operations, we elaborate that the operations in charge-qubit arrays are indeed robust against certain local perturbations.
In a weak link of two s-wave superconductors (SCs) coupled via a time-reversal-invariant (TRI) topological superconducting (TSC) island, a Josephson current can flow due to Cooper pairs tunneling in and out of spatially separated Majorana Kramers pairs (MKPs), which are doublets of Majorana bound states (MBSs). The sign of the resulting Josephson current is fixed by the joint parity of the four Majorana bound states that make up the MKPs on the TSC island. This parity-controlled Josephson effect can be used as a read-out mechanism for the joint parity in Majorana-based quantum computing. For a TSC island with four terminals, the SC leads can address a Majorana superconducting qubit (MSQ) formed by the charge ground states of the TSC island's terminals. Cooper pair splitting enables single-qubit operations, qubit read-out, as well as two-qubit entangling gates. Hence, TSC islands between SC leads may provide an alternative approach to superconducting quantum computation.
Superconducting circuits offer the opportunity to study quantum mechanics on mesoscopic scales unimpeded by dissipation. This fact and the nonlinearity of the Josephson inductance make it possible to use superconducting circuits as artificial atoms whose long-lived states can be selectively addressed and studied. A pronounced nonlinearity of the energy spectrum, however, requires quantum fluctuations of the flux across the Josephson junction which are large on the scale of the superconducting flux quantum ΦQ=h/2e. This implies charge fluctuations below the single Cooper-pair limit via flux-charge duality. The localization of charge leads to a strong susceptibility to interactions with charges in the environment which has motivated the search for schemes to decouple charges from their environment. This thesis is concerned with theoretical challenges arising from two complementary approaches to this problem: the realization of large impedances and the fractionalization of electrons by means of Majorana bound states.In recent years, the decoupling of charges from the environment through reactive large impedances, so-called ``superinductances'' L, has attracted much interest. These inductances feature small parasitic capacitance C such that the characteristic impedance L/C is much larger than the superconducting resistance quantum RQ=h/4e2. Superinductances have various applications ranging from qubit designs such as the 0-π qubit or the fluxonium to impedance matching, Bloch oscillations and the stabilization of phase slips in superconducting nanowires. Although there exists a well-established formalism for the quantization of superconducting circuits in terms of node fluxes, this formalism is ill-suited for the description of fast flux transport with localized charges in large-impedance environments. In particular, the nonlinear capacitive behavior of phase slip junctions cannot be modeled in a straightforward way using node fluxes. In view of the ever growing interest in superinductances, in the first part of the thesis, we present a recipe for quantization of planar circuits in terms of loop charges. As we will show, the loop charge approach is dual to the usual node flux formalism and well-adapted to a large impedance setting.In the second part of the thesis, we turn to a complementary approach of charge decoupling by means of Majorana bound states (MBS). MBS solve the decoupling problem by encoding a fermionic mode nonlocally into two bound states with large spatial separation such that a local coupling to the stored charge is no longer possible. It has been shown that despite the apparent nonlocality of the fermionic mode, transport through the mode remains local unless the MBS are coupled by a global perturbation like a finite charging energy. Here we show that, even in absence of charging energy, decoupling the superconductor from the ground plane achieves subtle coupling of the MBS that leads to nonlocal transport.Finally, in the last part of the thesis, we turn to two mesoscopic applications related to supersymmetric quantum mechanics. The simultaneous presence of a fermionic mode due to the MBS and a bosonic mode due to the Cooper-pair condensate makes systems involving MBS appealing candidates for the realization of supersymmetric quantum mechanics. For a Majorana Cooper-pair box, we discuss an unusual ``bosonic'' supersymmetry and its experimental signatures. Since MBS remain challenging to realize experimentally, we show that a similar supersymmetry can even be realized in a setup using standard superconducting circuitry without MBS.