We show how to exchange (braid) Majorana fermions in a network of superconducting nanowires by control over Coulomb interactions rather than tunneling. Even though Majorana fermions are charge-neutral quasiparticles (equal to their own antiparticle), they have an effective long-range interaction through the even-odd electron number dependence of the superconducting ground state. The flux through a split Josephson junction controls this interaction via the ratio of Josephson and charging energies, with exponential sensitivity. By switching the interaction on and off in neighboring segments of a Josephson junction array, the non-Abelian braiding statistics can be realized without the need to control tunnel couplings by gate electrodes.

Superconducting wires without time-reversal and spin-rotation symmetries can be driven into a topological phase that supports Majorana bound states. Direct detection of these zero-energy states is complicated by the proliferation of low-lying excitations in a disordered multimode wire. We show that the phase transition itself is signaled by a quantized thermal conductance and electrical shot noise power, irrespective of the degree of disorder. In a ring geometry, the phase transition is signaled by a period doubling of the magnetoconductance oscillations. These signatures directly follow from the identification of the sign of the determinant of the reflection matrix as a topological quantum number.

Proposals to measure non-Abelian anyons in a superconductor by quantum interference of vortices suffer from the predominantly classical dynamics of the normal core of an Abrikosov vortex. We show how to avoid this obstruction using coreless Josephson vortices, for which the quantum dynamics has been demonstrated experimentally. The interferometer is a flux qubit in a Josephson junction circuit, which can nondestructively read out a topological qubit stored in a pair of anyons -even though the Josephson vortices themselves are not anyons. The flux qubit does not couple to intra-vortex excitations, thereby removing the dominant restriction on the operating temperature of anyonic interferometry in superconductors.

The topological invariant of a topological insulator (or superconductor) is given by the number of symmetryprotected edge states present at the Fermi level. Despite this fact, established expressions for the topological invariant require knowledge of all states below the Fermi energy. Here we propose a way to calculate the topological invariant employing solely its scattering matrix at the Fermi level without knowledge of the full spectrum. Since the approach based on scattering matrices requires much less information than the Hamiltonianbased approaches (surface versus bulk), it is numerically more efficient. In particular, is better suited for studying disordered systems. Moreover, it directly connects the topological invariant to transport properties potentially providing a new way to probe topological phases.

The topological quantum number Q of a superconducting or chiral insulating wire counts the number of stable bound states at the end points. We determine Q from the matrix r of reflection amplitudes from one of the ends, generalizing the known result in the absence of time-reversal and chiral symmetry to all five topologically nontrivial symmetry classes. The formula takes the form of the determinant, Pfaffian, or matrix signature of r, depending on whether r is a real matrix, a real antisymmetric matrix, or a Hermitian matrix. We apply this formula to calculate the topological quantum number of N coupled dimerized polymer chains, including the effects of disorder in the hopping constants. The scattering theory relates a topological phase transition to a conductance peak, of quantized height and with a universal (symmetry class independent) line shape. Two peaks which merge are annihilated in the superconducting symmetry classes, while they reinforce each other in the chiral symmetry classes.

Qubits constructed from uncoupled Majorana fermions are protected from decoherence, but to perform a quantum computation this topological protection needs to be broken. Parity-protected quantum computation breaks the protection in a minimally invasive way, by coupling directly to the fermion parity of the system -irrespective of any quasiparticle excitations. Here we propose to use a superconducting charge qubit in a transmission line resonator (a socalled transmon) to perform parity-protected rotations and read-out of a topological (top) qubit. The advantage over an earlier proposal using a flux qubit is that the coupling can be switched on and off with exponential accuracy, promising a reduced sensitivity to charge noise.

We analyze the current-biased Shapiro experiment in a Josephson junction formed by two one-dimensional nanowires featuring Majorana fermions. Ideally, these junctions are predicted to have an unconventional $4\pi$-periodic Josephson effect and thus only Shapiro steps at even multiples of the driving frequency. Taking additionally into account overlap between the Majorana fermions, due to the finite length of the wire, renders the Josephson junction conventional for any dc-experiments. We show that probing the current-phase relation in a current biased setup dynamically decouples the Majorana fermions. We find that besides the even integer Shapiro steps there are additional steps at odd and fractional values. However, different from the voltage biased case, the even steps dominate for a wide range of parameters even in the case of multiple modes thus giving a clear experimental signature of the presence of Majorana fermions.Comment: 5+10pages, 5+8 Figures, published in PRB (rapid

The Josephson energy of two superconducting islands containing Majorana fermions is a 4πperiodic function of the superconducting phase difference. If the islands have a small capacitance, their ground state energy is governed by the competition of Josephson and charging energies. We calculate this ground state energy in a ring geometry, as a function of the flux Φ enclosed by the ring, and show that the dependence on the Aharonov-Bohm phase 2eΦ/ remains 4π-periodic regardless of the ratio of charging and Josephson energies -provided that the entire ring is in a topologically nontrivial state. If part of the ring is topologically trivial, then the charging energy induces quantum phase slips that restore the usual 2π-periodicity.