Roadmap to Majorana surface codes

Surface codes offer a very promising avenue towards fault-tolerant quantum computation. We argue that two-dimensional interacting networks of Majorana bound states in topological superconductor/semiconductor heterostructures hold several distinct advantages in that direction, both concerning the hardware realization and the actual operation of the code. We here discuss how topologically protected logical qubits in this Majorana surface code architecture can be defined, initialized, manipulated, and read out. All physical ingredients needed to implement these operations are routinely used in topologically trivial quantum devices. In particular, we show that by means of quantum interference terms in linear conductance measurements, composite single-electron pumping protocols, and gate-tunable tunnel barriers, the full set of quantum gates required for universal quantum computation can be implemented.Comment: 23 pages, 8 figure

Towards realistic implementations of a Majorana surface code

Surface codes have emerged as promising candidates for quantum information processing. Building on the previous idea to realize the physical qubits of such systems in terms of Majorana bound states supported by topological semiconductor nanowires, we show that the basic code operations, namely projective stabilizer measurements and qubit manipulations, can be implemented by conventional tunnel conductance probes and charge pumping via single-electron transistors, respectively. The simplicity of the access scheme suggests that a functional code might be in close experimental reach.Comment: 5 pages, 1 p. suppl.mat, PRL in pres

Comment on ``Density of States of Disordered Two-Dimensional Crystals with Half-Filled Band''

In a recent letter (PRL 84, 3930 (2000)) Nakhmedov et al. claimed that the Van Hove singularity at $\epsilon=0$ in the density of states (DoS) of the two-dimensional crystal with half-filled tight-binding band survives the addition of substitutional impurities. This derivation suffers from several inconsistencies. We resolve them and show that the DoS at the band center is finite, as one might naively expect

Crossovers between superconducting symmetry classes

We study the average density of states in a small metallic grain coupled to two superconductors with the phase difference $\pi$, in a magnetic field. The spectrum of the low-energy excitations in the grain is described by the random matrix theory whose symmetry depends on the magnetic field strength and coupling to the superconductors. In the limiting cases, a pure superconducting symmetry class is realized. For intermediate magnetic fields or couplings to the superconductors, the system experiences a crossover between different symmetry classes. With the help of the supersymmetric sigma-model we derive the exact expressions for the average density of states in the crossovers between the symmetry classes A-C and CI-C.Comment: 6 page

Level statistics inside the core of a superconductive vortex

Microscopic theory of the type of Efetov's supermatrix sigma-model is constructed for the low-lying electron states in a mixed superconductive-normal system with disorder. The developed technique is used for the study of the localized states in the core of a vortex in a moderately clean superconductor (1/\Delta << \tau << 1/\omega_0 = E_F/\Delta^2). At sufficiently low energies E << \omega_{Th}, the energy level statistics is described by the "zero-dimensional" limit of this supermatrix theory, with the effective "Thouless energy" \omega_{Th} \sim (\omega_0/\tau)^{1/2}. Within this energy range the result for the density of states is equivalent to that obtained within Altland-Zirnbauer random matrix model of class C. Nonzero modes of the sigma-model increase the mean interlevel distance \omega_0 by the relative amount of the order of [2\ln(1/\omega_0\tau)]^{-1}.Comment: 5 pages, RevTeX. One error is corrected, also two references are added. Submitted to JETP Letter

Field Theory of the Random Flux Model

The long-range properties of the random flux model (lattice fermions hopping under the influence of maximally random link disorder) are shown to be described by a supersymmetric field theory of non-linear sigma model type, where the group GL(n|n) is the global invariant manifold. An extension to non-abelian generalizations of this model identifies connections to lattice QCD, Dirac fermions in a random gauge potential, and stochastic non-Hermitian operators.Comment: 4 pages, 1 eps figur

Mean Free Path and Energy Fluctuations in Quantum Chaotic Billiards

The elastic mean free path of carriers in a recently introduced model of quantum chaotic billiards in two and three dimensions is calculated. The model incorporates surface roughness at a microscopic scale by randomly choosing the atomic levels at the surface sites between -W/2 and W/2. Surface roughness yields a mean free path l that decreases as L/W^2 as W increases, L being the linear size of the system. But this diminution ceases when the surface layer begins to decouple from the bulk for large enough values of W, leaving more or less unperturbed states on the bulk. Consequently, the mean free path shows a minimum of about L/2 for W of the order of the band width. Energy fluctuations reflect the behavior of the mean free path. At small energy scales, strong level correlations manifest themselves by small values of the number of levels variance Sigma^2(E) that are close to Random Matrix Theory (RMT) in all cases. At larger energy scales, fluctuations are below the logarithmic behavior of RMT for l > L, and above RMT value when l < L.Comment: 8 twocolumn pages, seven figures, revtex and epsf macros. To be published in Physical Review B

Giant current fluctuations in an overheated single electron transistor

Interplay of cotunneling and single-electron tunneling in a thermally isolated single-electron transistor (SET) leads to peculiar overheating effects. In particular, there is an interesting crossover interval where the competition between cotunneling and single-electron tunneling changes to the dominance of the latter. In this interval, the current exhibits anomalous sensitivity to the effective electron temperature of the transistor island and its fluctuations. We present a detailed study of the current and temperature fluctuations at this interesting point. The methods implemented allow for a complete characterization of the distribution of the fluctuating quantities, well beyond the Gaussian approximation. We reveal and explore the parameter range where, for sufficiently small transistor islands, the current fluctuations become gigantic. In this regime, the optimal value of the current, its expectation value, and its standard deviation differ from each other by parametrically large factors. This situation is unique for transport in nanostructures and for electron transport in general. The origin of this spectacular effect is the exponential sensitivity of the current to the fluctuating effective temperature.Comment: 10 pages, 11 figure

Transition from Poisson to gaussian unitary statistics: The two-point correlation function

We consider the Rosenzweig-Porter model of random matrix which interpolates between Poisson and gaussian unitary statistics and compute exactly the two-point correlation function. Asymptotic formulas for this function are given near the Poisson and gaussian limit.Comment: 19 pages, no figure

Tail States in a Superconductor with Magnetic Impurities

A field theoretic approach is developed to investigate the profile and spectrum of sub-gap states in a superconductor subject to a weak magnetic impurity potential. Such states are found to be associated with inhomogeneous supersymmetry broken instanton configurations of the action.Comment: 4 pages, 2 eps figure