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.PACS numbers: 03.67. Pp, 03.67.Ac, 03.67.Lx, 74.78.Na Introduction.-In recent years, surface codes have established themselves as a potent platform for universal quantum information processing. The basic idea of a surface code (when operated as a so-called stabilizer code) is to implement few 'logical' qubits -the actual carriers of quantum information -via the correlation of a large number of physical qubits [1][2][3][4][5][6][7]. What at first sight appears to be a redundant scheme offers a number of powerful advantages: (i) a degree of tolerance to errors orders of magnitude higher than that of other approaches, (ii) the possibility to implement a code as a comparatively simple two-dimensional (2D) layout of cells coupled by nearest-neighbor interactions, and (iii) the fact that essential code operations, including error tracking without need of active error correction in Clifford operations and/or memory access, are controlled by classical software. (While non-local stabilizer codes may allow for even higher error thresholds [6], we here focus on local surface codes.) These are highly attractive features which, accordingly, come at a hefty price tag: a large number of physical qubits is required even for modest operations. In particular, logical non-Clifford gates (e.g., the T gate) are required for universality, whose faulttolerant implementation would require magic state distillation [6,8]. Under these conditions, Ref. [5] estimates that about 10 3−4 physical qubits are needed to encode a reasonably fault-tolerant information qubit, implying that about O(10 8 ) physical qubits are needed to run, say, serious factorization algorithms for integers with O(10 2 ) decimals. Achieving maximal simplicity in the implementation and in the access of individual qubits will therefore be a decisive factor in advancing from O(1) qubits to functional systems. In this Letter we argue that semiconductor Majorana hardware layouts offer striking and so far unnoticed advantages in this regard.

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

We study a charge two-channel Kondo model, demonstrating that recent experiments [Iftikhar et al, Nature 526, 233 (2015)] realize an essentially perfect quantum simulation -not just of its universal physics, but also nonuniversal effects away from the scaling limit. Numerical renormalization group calculations yield conductance lineshapes encoding RG flow to a critical point involving a free Majorana fermion. By mimicking the experimental protocol, the experimental curve is reproduced quantitatively over 9 orders of magnitude, although we show that far greater bandwidth/temperature separation is required to obtain the universal result. Fermi liquid instabilities are also studied: In particular, our exact analytic results for non-linear conductance provide predictions away from thermal equilibrium, in the regime of existing experiments.Introduction and results.-The Kondo effect provides a paradigmatic example for universality: systems with very different microscopic details exhibit the same behavior in terms of rescaled temperature T /T K , where the Kondo temperature T K is an emergent low-energy scale characteristic of the particular system [1].

We study the entanglement spectrum of topological systems hosting non-Abelian anyons. Akin to energy levels of a Hamiltonian, the entanglement spectrum is composed of symmetry multiplets. We find that the ratio between different eigenvalues within one multiplet is universal and is determined by the anyonic quantum dimensions. This result is a consequence of the conservation of the total topological charge. For systems with non-Abelian topological order, this generalizes known degeneracies of the entanglement spectrum, which are hallmarks of topological states. Experimental detection of these entanglement spectrum signatures may become possible in Majorana wires using multicopy schemes, allowing the measurement of quantum entanglement and its symmetry resolution.arXiv:1810.01853v2 [cond-mat.str-el]

We study a topological superconductor island with spatially separated Majorana modes coupled to multiple normal-metal leads by single-electron tunneling in the Coulomb blockade regime. We show that low-temperature transport in such a Majorana island is carried by an emergent charge-e boson composed of a Majorana mode and an electronic excitation in leads. This transmutation from Fermi to Bose statistics has remarkable consequences. For noninteracting leads, the system flows to a non-Fermi-liquid fixed point, which is stable against tunnel couplings anisotropy or detuning away from the charge-degeneracy point. As a result, the system exhibits a universal conductance at zero temperature, which is a fraction of the conductance quantum, and low-temperature corrections with a universal power-law exponent. In addition, we consider Majorana islands connected to interacting onedimensional leads, and find different stable fixed points near and far from the charge-degeneracy point. DOI: 10.1103/PhysRevB.96.205403 Majorana modes are an unusual type of quasiparticles in topological superconductors, consisting of localized electron and hole excitations in an equal superposition [1][2][3]. The presence of spatially separated Majorana modes in a macroscopic topological superconductor gives rise to degenerate ground states that are locally indistinguishable and topologically protected. In a mesoscopic superconductor island with Majoranas (a Majorana island), however, these ground states partially split into two charge-parity sectors with the total number of electrons being even and odd, respectively; this energy splitting is unrelated to Majorana mode hybridization, but comes from the charging energy and can be tuned by a gate voltage [4,5]. This tunability enables electric control of Majoranas as well as new schemes of braiding and quantum computation based on mesoscopic topological superconductor devices [6][7][8][9][10][11][12][13].The interplay between Majorana modes and charging energy gives rise to a variety of topological quantum phenomena at the mesoscopic scale. One example is transport through a topological superconductor island with two spatially separated Majorana modes, each connected to a normal-metal lead by electron tunneling [4,14,15]. Theory [4] predicts that an unusual resonant tunneling process involving two distant Majoranas gives rise to a phase-coherent charge-e transport dubbed electron teleportation, exhibiting a conductance peak when the island is at a charge-degeneracy point. In a recent groundbreaking experiment [16] on proximitized nanowires under a magnetic field-a promising platform for topological superconductivity [17][18][19][20]-1e-periodic zero-bias conductance through the superconducting island has been observed in the Coulomb blockade regime, providing experimental support for electron teleportation via Majorana modes.In this paper, we study multiterminal charge transport through a Majorana island connected with M > 2 leads, each tunnel coupled to a Majorana zero mode, as shown in Fig. 1...

The phenomenon of charge fractionalization describes the emergence of novel excitations with fractional quantum numbers, as predicted in strongly correlated systems such as spin liquids. We elucidate that precisely such an unusual effect may occur in the simplest possible non-Fermi liquid, the two-channel Kondo effect. To bring this concept down to experimental test, we study nonequilibrium transport through a device realizing the charge two-channel Kondo critical point in a recent experiment by Iftikhar et al. [Nature (London) 526, 233 (2015)NATUAS0028-083610.1038/nature15384]. The shot noise at low voltages is predicted to result in a universal Fano factor e^{*}/e=1/2. This allows us to experimentally identify elementary transport processes of emergent fermions carrying half-integer charge.

We study a Majorana island coupled to a bulk superconductor via a Josephson junction and to multiple external normal leads. In the absence of the Josephson coupling, the system displays a topological Kondo state, which had been largely studied recently. However, we find that this state is unstable even to small Josephson coupling, which instead leads at low temperature T to a new fixed point. Most interesting is the case of three external leads, forming a minimal electronic realization of the long sought two-channel Kondo effect. While the T = 0 conductance corresponds to simple resonant Andreev reflection, the leading T dependence forms an experimental fingerprint for non-Fermi liquid properties.