Majorana fermions hold promise for quantum computation, because their non-Abelian braiding statistics allows for topologically protected operations on quantum information. Topological qubits can be constructed from pairs of well-separated Majoranas in networks of nanowires. The coupling to a superconducting charge qubit in a transmission line resonator (transmon) permits braiding of Majoranas by external variation of magnetic fluxes. We show that readout operations can also be fully flux-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 efficient creation of highly entangled states and simplify quantum error correction protocols by avoiding the need for ancilla qubits.After the first signatures were reported [1][2][3][4] of Majorana bound states in superconducting nanowires [5][6][7], the quest for non-Abelian braiding statistics [8][9][10][11] has intensified. Much interest towards Majorana fermions arises from their technological potential in fault-tolerant quantum computation [12][13][14][15][16]. Their non-Abelian exchange statistics would allow to perform quantum gates belonging to the Clifford group with extremely good accuracy. Moreover, topological qubits encoded non-locally in well-separated Majorana bound states would be resilient against many sources of decoherence. Even without the applications in quantum information processing, observing a new type of quantum statistics would be a milestone in the history of physics.The two central issues for the application of Majorana fermions are (i) how to unambiguously demonstrate their non-Abelian exchange statistics and (ii) how to exploit their full potential for quantum information processing. The first issue requires an elementary circuit that can perform three tasks: initialization of a qubit, braiding (exchange) of two Majoranas, and finally measurement (readout) of the qubit. In view of the second issue, this circuit should be scalable and serve as a first step towards universal fault-tolerant quantum computation.Here we present such a circuit, using a superconducting charge qubit in a transmission line resonator (transmon [17][18][19][20]) to initialize, control, and measure the topological qubit. In such a hybrid system, named top-transmon [21], the long-range Coulomb couplings of Majorana fermions can be used to braid them and to read out their fermion parity [21,22]. While there exist several proposals to control or measure Majorana fermions in nanowires [11,[21][22][23][24][25][26][27][28][29][30][31][32], combining braiding and measurement without local adjustment of microscopic parameters remains a challenge. We show that full macroscopic control is possible if during the...

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.

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. DOI: 10.1103/PhysRevLett.115.127002 PACS numbers: 85.25.Hv, 62.23.Hj, 74.45.+c, 84.40.Dc In superconducting circuits, macroscopic degrees of freedom like currents and voltages can exhibit quantum mechanical behavior. These circuits can behave as artificial atoms with discrete, anharmonic levels whose transitions can be driven coherently [1]. In the field of circuit quantum electrodynamics (cQED), these artificial atoms are coupled to resonators to perform microwave quantum optics in the solid state [2,3]. Over the past decade, cQED has also grown into a promising platform for quantum information processing, wherein the ground and first-excited levels of each atom serve as an effective qubit [4]. To date, implementations of superconducting quantum circuits have relied almost exclusively on aluminum-aluminum-oxidealuminum (Al=AlOx=Al) tunnel junctions as the source of nonlinearity without dissipation. However, many exciting applications require magnetic fields (∼0.5 T) at which superconductivity in aluminum is destroyed, calling for an alternative approach to realizing microwave artificial atoms.Recent advances in materials development and nanowire (NW) growth have enabled the development of superconductor-semiconductor structures supporting coherent charge transport without dissipation [5] and providing signatures of Majorana bound states [6]. To date, superconductor-semiconductor-superconductor Josephson elements (JEs) have been studied exclusively in quasi-dc [7][8][9][10] and radio-frequency [11] transport. Building microwave circuits operating in the quantum regime, in which transition energies between levels exceed the thermal energy, offers new ways to investigate the physics of hybrid superconductor-semiconductor structures using spectroscopy [12][13][14][15].In this Letter, we report the realization of microwavefrequency cQED circuits made from hybrid JEs based on InAs NWs contacted by NbTiN. Capacitively shunted single JEs behave as weakly anharmonic oscillators, or transmons [16], with transition frequencies tunable by the field effect, i.e., voltage on a proximal side gate. Doubleelement devices show similar transmonlike behavior at zero applied flux but behave as flux qubits [17] near full frustration owing to a double-well Josephson potential arising from nonsinusoidal current-phase relations (CPR s). We observe microwave-driven transitions between states ...

We present a novel route to realizing topological superconductivity using magnetic flux applied to a full superconducting shell surrounding a semiconducting nanowire core. In the destructive Little-Parks regime, reentrant regions of superconductivity are associated with integer number of phase windings in the shell. Tunneling into the core reveals a hard induced gap near zero applied flux, corresponding to zero phase winding, and a gapped region with a discrete zero-energy state around one applied flux quantum, Φ0 = h/2e, corresponding to 2π phase winding. Theoretical analysis indicates that in the presence of radial spin-orbit coupling in the semiconductor, the winding of the superconducting phase can induce a transition to a topological phase supporting Majorana zero modes. Realistic modeling shows a topological phase persisting over a wide range of parameters, and reproduces experimental tunneling conductance data. Further measurements of Coulomb blockade peak spacing around one flux quantum in full-shell nanowire islands shows exponentially decreasing deviation from 1e periodicity with device length, consistent with Majorana modes at the ends of the nanowire. arXiv:2003.13177v1 [cond-mat.mes-hall]

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 Ref.1 , and identifies the signatures of the topological transition in the two-terminal conductance.

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.

AUTHOR CONTRIBUTIONS D. J. v. W., A. P. and D. B. performed the experiments. B. v. H., J. I. V. and L. I. G. developed the theory to analyze the data. P. K. and J. N. contributed to the nanowire growth. D. J. v. W., A. P. and D. B. fabricated the samples. L. P. K. and A. G. designed and supervised the experiments. D. J. v. W.

We develop a unified numerical approach for modeling semiconductor-superconductor heterostructures. All the key physical ingredients of these systems -orbital effect of magnetic field, superconducting proximity effect and electrostatic environment -are taken into account on equal footing in a realistic device geometry. As a model system, we consider indium arsenide (InAs) nanowires with epitaxial aluminum (Al) shell, which is one of the most promising platforms for Majorana zero modes. We demonstrate qualitative and quantitative agreement of the obtained results with the existing experimental data. Finally, we characterize the topological superconducting phase emerging in a finite magnetic field and calculate the corresponding topological phase diagram. arXiv:1810.04180v2 [cond-mat.supr-con]