B. van Heck

Flux-controlled quantum computation with Majorana fermions

uxes. We show that readout operations can also be fully ux-controlled, without requiring microscopic control over tunnel couplings. We identify the minimal circuit that can perform the initialization{braiding{measurement steps required to demonstrate non-Abelian statistics. We introduce the Random Access Majorana Memory, a scalable circuit that can perform a joint parity measurement on Majoranas belonging to a selection of topological qubits. Such multi-qubit measurements allow for the ecient creation of highly entangled states and simplify quantum error correction protocols by avoiding the need for ancilla qubits.

Coulomb-assisted braiding of Majorana fermions in a Josephson junction array

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.

Realization of Microwave Quantum Circuits Using Hybrid Superconducting-Semiconducting Nanowire Josephson Elements

We report the realization of quantum microwave circuits using hybrid superconductor-semiconductor Josephson elements comprised of InAs nanowires contacted by NbTiN. Capacitively shunted single elements behave as transmon circuits with electrically tunable transition frequencies. Two-element circuits also exhibit transmonlike behavior near zero applied flux but behave as flux qubits at half the flux quantum, where nonsinusoidal current-phase relations in the elements produce a double-well Josephson potential. These hybrid Josephson elements are promising for applications requiring microwave superconducting circuits operating in a magnetic field.

Coulomb stability of the 4pi-periodic Josephson effect of Majorana fermions

The Josephson energy of two superconducting islands containing Majorana fermions is a 4\pi-periodic 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 -\Phi- enclosed by the ring, and show that the dependence on the Aharonov-Bohm phase 2e\Phi/\hbar remains 4\pi-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 2\pi-periodicity.

Statistical Topological Insulators

We define a class of insulators with gapless surface states protected from localization due to the statistical properties of a disordered ensemble, namely due to the ensembles invariance under a certain symmetry. We show that these insulators are topological, and are protected by a

Conductance of a proximitized nanowire in the Coulomb blockade regime

\mathbb{Z}_2

Single fermion manipulation via superconducting phase differences in multiterminal Josephson junctions

invariant. Finally, we prove that every topological insulator gives rise to an infinite number of classes of statistical topological insulators in higher dimensions. Our conclusions are confirmed by numerical simulations.

Topological phases in two-dimensional arrays of parafermionic zero modes

We identify the leading processes of electron transport across finite-length segments of proximitized nanowires and build a quantitative theory of their two-terminal conductance. In the presence of spin-orbit interaction, a nanowire can be tuned across the topological transition point by an applied magnetic field. Due to a finite segment length, electron transport is controlled by the Coulomb blockade. Upon increasing of the field, the shape and magnitude of the Coulomb blockade peaks in the linear conductance is defined, respectively, by Andreev reflection, single-electron tunneling, and resonant tunneling through the Majorana modes emerging after the topological transition. Our theory provides the framework for the analysis of experiments with proximitized nanowires, such as reported in Albrecht et al., Nature 531, 206-209 (2016), and identifies the signatures of the topological transition in the two-terminal conductance.

Braiding of non-Abelian anyons using pairwise interactions.

We show how the superconducting phase difference in a Josephson junction may be used to split the Kramers degeneracy of its energy levels and to remove all the properties associated with time-reversal symmetry. The superconducting phase difference is known to be ineffective in two-terminal short Josephson junctions, where irrespective of the junction structure the induced Kramers degeneracy splitting is suppressed and the ground state fermion parity must stay even, so that a protected zero-energy Andreev level crossing may never appear. Our main result is that these limitations can be completely avoided by using multiterminal Josephson junctions. There the Kramers degeneracy breaking becomes comparable to the superconducting gap, and applying phase differences may cause the change of the ground state fermion parity from even to odd. We prove that the necessary condition for the appearance of a fermion parity switch is the presence of a “discrete vortex” in the junction: the situation when the phases of the superconducting leads wind by 2?. Our approach offers strategies for creation of Majorana bound states as well as spin manipulation. Our proposal can be implemented using any low density, high spin-orbit material such as InAs quantum wells, and can be detected using standard tool.

Zeeman and spin-orbit effects in the Andreev spectra of nanowire junctions

It has recently been realized that zero modes with projective non-Abelian statistics, generalizing the notion of Majorana bound states, may exist at the interface between a superconductor and a ferromagnet along the edge of a fractional topological insulator (FTI). Here, we study two-dimensional architectures of these non-Abelian zero modes, whose interactions are generated by the charging and Josephson energies of the superconductors. We derive low-energy Hamiltonians for two different arrays of FTIs on the plane, revealing an interesting interplay between the real-space geometry of the system and its topological properties. On the one hand, in a geometry where the length of the FTI edges is independent on the system size, the array has a topologically ordered phase, giving rise to a qudit toric code Hamiltonian in perturbation theory. On the other hand, in a geometry where the length of the edges scales with system size, we find an exact duality to an Abelian lattice gauge theory and no topological order.