We theoretically propose a set of universal quantum gates acting on a hybrid qubit formed by coupling a quantum-dot spin qubit and Majorana fermion qubit. First, we consider a quantum dot that is tunnel coupled to two topological superconductors. The effective spin-Majorana exchange facilitates a hybrid CNOT gate for which either qubit can be the control or target. The second setup is a modular scalable network of topological superconductors and quantum dots. As a result of the exchange interaction between adjacent spin qubits, a CNOT gate is implemented that acts on neighboring Majorana qubits and eliminates the necessity of interqubit braiding. In both setups, the spin-Majorana exchange interaction allows for a phase gate, acting on either the spin or the Majorana qubit, and for a SWAP or hybrid SWAP gate which is sufficient for universal quantum computation without projective measurements.
We study two microscopic models of topological insulators in contact with an s-wave superconductor. In the first model the superconductor and the topological insulator are tunnel coupled via a layer of scalar and of randomly oriented spin impurities. Here, we require that spin-flip tunneling dominates over spin-conserving one. In the second model the tunnel coupling is realized by an array of single-level quantum dots with randomly oriented spins. It is shown that the tunnel region forms a π-junction where the effective order parameter changes sign. Interestingly, due to the random spin orientation the effective descriptions of both models exhibit time-reversal symmetry. We then discuss how the proposed π-junctions support topological superconductivity without magnetic fields and can be used to generate and manipulate Kramers pairs of Majorana fermions by gates. Introduction. When two s-wave superconductors (SCs) are brought into contact via an insulator the energy of the system in equilibrium is minimized when the relative phase between the two superconducting order parameters vanishes. Interestingly, when the insulator is doped with magnetic impurities, it was shown by theory [1] and experiment [2] that spin-flip tunneling can induce an equilibrium ground state with a relative phase difference of π between the superconducting order parameters, building up a so-called Josephson π-junction (JπJ). It was predicted [3] and experimentally confirmed [4] that a JπJ can be generated by replacing the layer of magnetic impurities by a ferromagnetic metal. A JπJ can also arise when two SCs are tunnel-coupled through an intermediate resonant state in the presence of strong Coulomb interactions [5], as observed in a system of two SCs coupled by a quantum dot (QD) occupied by a single electron [6]. In recent experiments [7][8][9] it was demonstrated that superconductivity can also be proximity-induced in the helical edge states of a topological insulator (TI) material [10][11][12][13][14][15][16][17][18] via coupling to an external s-wave SC. These experimental advances have also stimulated the theoretical interest in Josephson junctions based on TIs [19][20][21]. Motivated by the existence of ordinary JπJs an important and immediate question is: Are there microscopic mechanisms allowing one to induce a superconducting order parameter in the helical edge states of the TI that is of opposite relative sign compared to the one of the external s-wave SC, ideally without breaking time-reversal invariance (TRI)? In this work we answer this question in the affirmative.
We find a class of topological superconductors which possess an emergent time-reversal symmetry that is present only after projecting to an effective low-dimensional model. We show that a topological phase in symmetry class DIII can be realized in a noninteracting system coupled to an s-wave superconductor only if the physical time-reversal symmetry of the system is broken, and we provide three general criteria that must be satisfied in order to have such a phase. We also provide an explicit model which realizes the class DIII topological superconductor in 1D. We show that, just as in time-reversal invariant topological superconductors, the topological phase is characterized by a Kramers pair of Majorana fermions that are protected by the emergent time-reversal symmetry. DOI: 10.1103/PhysRevB.96.161407 Introduction. Topological superconductors have been intensively pursued in recent years [1][2][3] because the Majorana fermions which are localized to their boundaries have potential applications in the development of a topological quantum computer [4,5]. The most promising proposals to date for engineering topological superconductivity involve coupling a conventional superconductor either to a nanowire with Rashba spin-orbit interaction that is subjected to an external magnetic field or to a ferromagnetic atomic chain [27][28][29][30][31][32][33][34].Additionally, there have been several proposals to engineer topological superconductors in symmetry class DIII. Such systems possess both particle-hole symmetry and time-reversal symmetry [35], with the presence of time-reversal symmetry ensuring that the Majorana fermions existing at the boundaries of class DIII topological superconductors come in Kramers pairs. In one dimension (1D), where superconductivity is required to be induced by the proximity effect, it has been shown that a nontrivial topological phase in class DIII can be realized by proximity coupling a noninteracting multichannel Rashba nanowire to an unconventional superconductor [36][37][38][39] or to two conventional superconductors forming a Josephson junction with a phase difference of π [40]. Alternatively, an effective π -phase difference can be induced in a multichannel Rashba nanowire with repulsive electron-electron interactions [41] or in a system of two topological insulators coupled to a conventional superconductor via a magnetic insulator [42]. It has also been proposed to realize class DIII topological superconductivity in a system of two Rashba nanowires [43][44][45] or two topological insulators [46] coupled to a single conventional superconductor, but repulsive interactions are also necessary to reach the topological phase in these setups, which require a strength of induced crossed Andreev (interwire) pairing exceeding that of the direct (intrawire) pairing [47][48][49]. While it would be beneficial to engineer a DIII topological superconductor in a noninteracting 1D system coupled to a single conventional superconductor, as such a setup could avoid relying on unconventional superc...
A hard proximity-induced superconducting gap has recently been observed in semiconductor nanowire systems at low magnetic fields. However, in the topological regime at high magnetic fields, a soft gap emerges and represents a fundamental obstacle to topologically protected quantum information processing with Majorana bound states. Here we show that in a setup of double Rashba nanowires that are coupled to an s-wave superconductor and subjected to an external magnetic field along the wires, the topological threshold can be significantly reduced by the destructive interference of direct and crossed-Andreev pairing in this setup, precisely down to the magnetic field regime in which current experimental technology allows for a hard superconducting gap. We also show that the resulting Majorana bound states exhibit sufficiently short localization lengths, which makes them ideal candidates for future braiding experiments.
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