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We report tunneling spectroscopy and transport measurements in superconducting Al and ferromagnetic-insulator EuS bilayers. The samples display remanent spin-splitting, roughly half the superconducting gap, and supercurrent transport above the average paramagnetic limit. We interpret this behavior as arising from the interplay between two characteristic length scales: the superconducting coherence length, $\xi$, and the magnetic domain size, $d$. By comparing experimental results to a theoretical model, we find $\xi/d \approx 10$. In this regime, spin-averaging across the micromagnetic configuration can locally suppress superconductivity, resulting in percolative supercurrent flow., Comment: 15 pages, 4+7 figures

Arrays of superconducting qubits and cavities offer a promising route for realizing highly controllable artificial materials. However, many analog simulations of superconducting circuit hardware have focused on bosonic systems. Fermionic simulations, on the other hand, have largely relied on digital approaches that require non-local qubit couplings, which could limit their scalability. Here, we propose and study an alternative approach for analog fermionic quantum simulation based on arrays of coherently coupled mesoscopic Josephson junctions. These Josephson junction arrays implement an effective superlattice of Andreev bound state "atoms" that can trap individual fermionic quasiparticles and, due to their wavefunction overlap, mediate quasiparticle hoppings. By developing a Wannier function approach, we show that these Andreev bound state arrays form an all-superconducting and circuit QED-compatible platform for emulating lattice models of fermionic quasiparticles that are phase- and gate-programmable. Interestingly, we also find that the junction lattices can undergo a topological transition and host fermionic boundary modes that can be probed by conductance measurements. We hope our results will inspire the realization of artificial and possibly topological materials on Andreev bound state quantum simulators.

Josephson diodes are superconducting elements that show an asymmetry in the critical current depending on the direction of the current. Here, we theoretically explore how an alternating current bias can tune the response of such a diode. We show that for slow driving there is always a regime where the system can only carry zero-voltage dc current in one direction, thus effectively behaving as an ideal Josephson diode. Under fast driving, the diode efficiency is also tunable, although the ideal regime cannot be reached in this case. We also investigate the residual dissipation due to the time-dependent current bias and show that it remains small. All our conclusions are solely based on the critical current asymmetry of the junction, and are thus compatible with any Josephson diode., Comment: 4 pages, 4 figures (+ 9 pages, 3 figures)

Dissipationless nonlinearities for three-wave mixing are a key component of many superconducting quantum devices, such as amplifiers and bosonic qubits. So far, such third-order nonlinearities have been primarily achieved with circuits of concatenated Josephson tunnel junctions. In this work, we theoretically develop an alternative approach to realize third-order nonlinearities from gate-tunable and intrinsically symmetry-broken quantum material Josephson junctions. We illustrate this approach on two examples, an Andreev interferometer and a magnetic Josephson junction. Our results show that both setups enable Kerr-free three-wave mixing for a broad range of frequencies, an attribute that is highly desirable for amplifier applications. Moreover, we also find that the magnetic junction constitutes a paradigmatic example for three-wave mixing in a minimal single-junction device without the need for any external biases. We hope that our work will guide the search of dissipationless nonlinearities in quantum material superconducting devices and inspire new ways of characterizing symmetry-breaking in quantum materials with microwave techniques.

Superconductor/semiconductor hybrid devices have attracted increasing interest in the past years. Superconducting electronics aims to complement semiconductor technology, while hybrid architectures are at the forefront of new ideas such as topological superconductivity and protected qubits. In this work, we engineer the induced superconductivity in two-dimensional germanium hole gas by varying the distance between the quantum well and the aluminum. We demonstrate a hard superconducting gap and realize an electrically and flux tunable superconducting diode using a superconducting quantum interference device (SQUID). This allows to tune the current phase relation (CPR), to a regime where single Cooper pair tunneling is suppressed, creating a $\sin \left( 2 \varphi \right)$ CPR. Shapiro experiments complement this interpretation and the microwave drive allows to create a diode with 100% efficiency. The reported results open up the path towards integration of spin qubit devices, microwave resonators and (protected) superconducting qubits on a silicon technology compatible platform.

Superconducting qubits with intrinsic noise protection offer a promising approach to improve the coherence of quantum information. Crucial to such protected qubits is the encoding of the logical quantum states into wavefunctions with disjoint support. Such encoding can be achieved by a Josephson element with an unusual charge-4e supercurrent emerging from the coherent transfer of pairs of Cooper-pairs. In this work, we demonstrate the controlled conversion of a conventional charge-2e dominated to a charge-4e dominated supercurrent in a superconducting quantum interference device (SQUID) consisting of gate-tunable planar Josephson junctions (JJs). We investigate the ac Josephson effect of the SQUID and measure a dominant photon emission at twice the fundamental Josephson frequency together with a doubling of the number of Shapiro steps, both consistent with the appearance of charge-4e supercurrent. Our results present a step towards novel protected superconducting qubits based on superconductor-semiconductor hybrid materials.

The Josephson diode (JD) is a non-reciprocal circuit element that supports a larger critical current in one direction compared to the other. This effect has gained a growing interest because of promising applications in superconducting electronic circuits with low power consumption. Some implementations of a JD rely on breaking the inversion symmetry in the material used to realize Josephson junctions (JJs), but two recent theoretical proposals have suggested that the effect can also be engineered by combining two JJs hosting highly transmitting Andreev bound states in a Superconducting Quantum Interference Device (SQUID) at a small, but finite flux bias [1, 2]. We have realized a SQUID with two JJs fabricated in a proximitized InAs two-dimensional electron gas (2DEG). We demonstrate gate control of the diode efficiency from zero up to around 30% at specific flux bias values which comes close to the maximum of approxiomately 40% predicated in Ref. [1]. The key ingredients to the JD effect in the SQUID arrangement is the presence of highly transmitting channels in the JJs, a flux bias and an asymmetry between the two SQUID arms., Comment: 14 pages, 9 figures (main text + appendix all in one file)

Coherent tunneling processes of multiple Cooper pairs across a Josephson junction give rise to higher harmonics in the current phase relation. In this work, we propose and study Josephson junctions based on semiconductor-superconductor-ferromagnetic insulator heterostructures to engineer nonsinusoidal current-phase relations. The gate-tunability of charge carriers density in the semiconductor, together with the adjustable magnetization of the ferromagnetic insulator, provides control over the content of the supercurrent harmonics. At finite exchange field, hybrid junctions can undergo a 0\,--\,$\pi$ phase transition, resulting in the supercurrent reversal. Close to the transition, single-pair tunneling is suppressed and the current-phase relation is dominated by the second-harmonic, indicating transport primarily by pairs of Cooper pairs. Finally, we demonstrate that non-collinear magnetization or spin-orbit coupling in the leads and the junction can lead to a gate-tunable Josephson diode effect with efficiencies of up to $\sim30\%$., Comment: 13 pages, 7+1 figures

A Josephson diode is a non-reciprocal circuit element that supports a larger dissipationless supercurrent in one direction than in the other. In this work, we propose and theoretically study a class of Josephson diodes based on supercurrent interferometers containing mesoscopic Josephson junctions, such as point contacts or quantum dots, which are not diodes themselves but possess non-sinusoidal current-phase relations. We show that such Josephson diodes have several important advantages, like being electrically tunable and requiring neither Zeeman splitting nor spin-orbit coupling, only time-reversal breaking by a magnetic flux. We also show that our diodes have a characteristic AC response, revealed by the Shapiro steps. Even the simplest realization of our Josephson diode paradigm that relies on only two junctions can achieve efficiencies of up to $\sim40\%$ and, interestingly, far greater efficiencies are achievable by concatenating multiple interferometer loops.

We report supercurrent transport measurements in hybrid Josephson junctions comprised of semiconducting InAs nanowires with epitaxial ferromagnetic insulator EuS and superconducting Al coatings. The wires display a hysteretic superconducting window close to the coercivity, away from zero external magnetic field. Using a multi-interferometer setup, we measure the current-phase relation of multiple magnetic junctions and find an abrupt switch between $\pi$ and 0 phases within the superconducting window. We attribute the 0-$\pi$ transition to the discrete flipping of the EuS domains and provide a qualitative theory showing that a sizable exchange field can polarize the junction and lead to the supercurrent reversal. Both $0$ and $\pi$ phases can be realized at zero external field by demagnetizing the wire.

Novel qubits with intrinsic noise protection constitute a promising route for improving the coherence of quantum information in superconducting circuits. However, many protected superconducting qubits exhibit relatively low transition frequencies, which could make their integration with conventional transmon circuits challenging. In this work, we propose and study a scheme for entangling a tunable transmon with a Cooper-pair parity-protected qubit, a paradigmatic example of a low-frequency protected qubit that stores quantum information in opposite Cooper-pair parity states on a superconducting island. By tuning the external flux on the transmon, we show that non-computational states can mediate a two-qubit entangling gate that preserves the Cooper-pair parity independent of the detailed pulse sequence. Interestingly, the entangling gate bears similarities to a controlled-phase gate in conventional transmon devices. Hence, our results suggest that standard high-precision gate calibration protocols could be repurposed for operating hybrid qubit devices.

We propose a protected qubit based on a modular array of superconducting islands connected by semiconductor Josephson interferometers. The individual interferometers realize effective $\cos2\phi$ elements that exchange `pairs of Cooper pairs' between the superconducting islands when gate-tuned into balance and frustrated by a half flux quantum. If a large capacitor shunts the ends of the array, the circuit forms a protected qubit because its degenerate ground states are robust to offset charge and magnetic field fluctuations for a sizable window around zero offset charge and half flux quantum. This protection window broadens upon increasing the number of interferometers if the individual elements are balanced. We use an effective spin model to describe the system and show that a quantum phase transition point sets the critical flux value at which protection is destroyed.

We introduce and study a realistic model for superconductivity in twisted bilayer WSe$_{2}$, where electron pairing arises from spin-valley fluctuations in the weak-coupling regime. Our model comprises both the full continuum model moir\'{e} bandstructure and a short-ranged repulsive interaction. By calculating the spin-valley susceptibility, we identify a Fermi surface nesting feature near half-filling of the top-most moir\'{e} band, which induces significantly enhanced spin-valley fluctuations. We then analyze the dominant Kohn-Luttinger pairing instabilities due to these spin-valley fluctuations and show that the leading instability corresponds to a two-component order parameter, which can give rise to nematic, chiral and topological superconductivity. As our findings are asymptotically exact for small interaction strengths, they provide a viable starting point for future studies of superconductivity in twisted transition metal dichalcogenide bilayers.

High entropy alloy(HEA) films of CrFeCoNiCu were prepared by sputtering, their structure was characterized, and their electric properties measured by temperature dependent Hall and Seebeck measurement. The HEA films show a solid solution with fcc structure, and a 111 texture with columnar grains of widths 15-30 nm extending through film thickness with very many twins. The residual electrical resistivity of the films is around 140 {\mu}{\Omega}cm and the temperature dependence of the resistivity is metal-like. The temperature coefficient of resistivity (TCR) is small (2 ppm/K). The Hall coefficient is positive while the Seebeck coefficients is negative. This is interpreted as arising from an electronic structure where the Fermi level passes through band states having both holes and electrons as indicated by band structure calculations. The HEA structure appears stable for annealing in vacuum, while annealing in an oxygen containing atmosphere causes the surface to oxidize and grow a Cr-rich oxide on the surface. This is then accompanied by demixing of the HEA solid solution and a decrease in residual resistance of the film., Comment: Manuscript, 30 pages double spaced 8 figures, Accepted manuscript by Journal of Vacuum Science and Technology A 2021

We study a Cooper-pair transistor realized by two Josephson weak links that enclose a superconducting island in an InSb-Al hybrid nanowire. When the nanowire is subject to a magnetic field, isolated subgap levels arise in the superconducting island and, due to the Coulomb blockade,mediate a supercurrent by coherent co-tunneling of Cooper pairs. We show that the supercurrent resulting from such co-tunneling events exhibits, for low to moderate magnetic fields, a phase offset that discriminates even and odd charge ground states on the superconducting island. Notably,this phase offset persists when a subgap state approaches zero energy and, based on theoretical considerations, permits parity measurements of subgap states by supercurrent interferometry. Such supercurrent parity measurements could, in a new series of experiments, provide an alternative approach for manipulating and protecting quantum information stored in the isolated subgap levels of superconducting islands.

We prove general adjoint functor theorems for weakly (co)complete $n$-categories. This class of $n$-categories includes the homotopy $n$-categories of (co)complete $\infty$-categories, so these $n$-categories do not admit all small (co)limits in general. We also introduce Brown representability for (homotopy) $n$-categories and prove a Brown representability theorem for localizations of compactly generated $n$-categories. This class of $n$-categories includes the homotopy $n$-categories of presentable $\infty$-categories if $n \geq 2$, and the homotopy $n$-categories of presentable stable $\infty$-categories for any $n \geq 1$., Comment: n-GAFT is shown also for n=2; corrected some errors around the proof of Lemma 3.2.9 (Criterion B) and the corresponding examples; other minor improvements in the exposition and some more details. To appear in Documenta Mathematica

ZrNiSn and related half Heusler compounds are candidate materials for efficient thermoelectric energy conversion with a reported thermoelectric figure-of-merit of n-type ZrNiSn exceeding unity. Progress on p-type materials has been more limited, which has been attributed to the presence of an impurity band, possibly related to the presence of Ni interstitials in nominally vacant 4d position. The specific energetic position of this band, however, has not been resolved. Here, we report results of a concerted theory-experiment investigation for a nominally undoped ZrNiSn, based on measurements of electrical resistivity, Hall coefficient, Seebeck coefficient and Nernst coefficient, measured in a temperature range from 80 to 420 K. The results are analyzed with a semi-analytical model combining a density functional theory (DFT) description for ideal ZrNiSn, with a simple analytical correction for the impurity band. The model provides a good quantitative agreement with experiment, describing all salient features in the full temperature span for the Hall, conductivity, and Seebeck measurements, while also reproducing key trends in the Nernst results. This comparison pinpoints the impurity band edge to 40 meV below the conduction band edge, which agrees well with a separate DFT study of a supercell containing Ni interstitials. Moreover, we corroborate our result with a separate study of ZrNiSn0.9Pb0.1 sample showing similar agreement with an impurity band edge shifted to 32 meV below the conduction band.

Spinal cord injuries frequently impair the ability to walk. Powered lower limb exoskeletons offer a promising solution to restore walking ability. However, they are currently restricted to even ground. We hypothesized that compliant exoskeleton knees could decrease required effort to maneuver on uneven terrain, and increase gait velocity and stability. We describe a case study of a motor-complete spinal cord injury user (AIS A, Th12) walking with a powered exoskeleton on even and uneven ground over multiple sessions after extensive training. Measurements with compliant or rigid exoskeleton knee joints were performed on three different days for each configuration. Body motion and crutch ground interaction forces were recorded to assess gait performance. We observed higher walking speeds with a compliant exoskeleton knee configuration (mean: 0.116 m/s on uneven and 0.145 m/s on even ground) compared to a rigid configuration (mean: 0.083 m/s and 0.100 m/s). Crutch force impulse was significantly reduced in the compliant configuration. Lastly, gait was more symmetric when the knee joints were compliant. In conclusion, compliant exoskeleton knee joints can help maneuver uneven ground faster and with less user effort than rigid joints. Based on our findings, exoskeleton designers should consider introducing compliance into their design to increase gait robustness and performance, and render exoskeletons more suitable for daily life use., Comment: 18 pages, 7 figures, 3 tables

Powered lower limb exoskeletons are a viable solution for people with a spinal cord injury to regain mobility for their daily activities. However, the commonly employed rigid actuation and pre-programmed trajectories increase the risk of falling in case of collisions with external objects. Compliant actuation may reduce forces during collisions, thus protecting hardware and user. However, experimental data of collisions specific to lower limb exoskeletons are not available. In this work, we investigated how a variable stiffness actuator at the knee joint influences collision forces transmitted to the user via the exoskeleton. In a test bench experiment, we compared three configurations of an exoskeleton leg with a variable stiffness knee actuator in (i) compliant or (ii) stiff configurations, and with (iii) a rigid actuator. The peak torque observed at the pelvis was reduced from 260.2 Nm to 116.2 Nm as stiffness decreased. In addition, the mechanical impulse was reduced by a factor of three. These results indicate that compliance in the knee joint of an exoskeleton can be favorable in case of collision and should be considered when designing powered lower limb exoskeletons. Overall, this could decrease the effort necessary to maintain balance after a collision and improved collision handling in exoskeletons could result in safer use and benefit their usefulness in daily life., Comment: 9 pages, 7 figures, 3 tabels

We introduce and study a minimum two-orbital Hubbard model on a triangular lattice, which captures the key features of both the trilayer ABC-stacked graphene-boron nitride heterostructure and twisted transition metal dichalcogenides in a broad parameter range. Our model comprises first- and second-nearest neighbor hoppings with valley-contrasting flux that accounts for trigonal warping in the band structure. For the strong-coupling regime with one electron per site, we derive a spin-orbital exchange Hamiltonian and find the semiclassical ground state to be a spin-valley density wave. We show that a relatively small second-neighbor exchange interaction is sufficient to stabilize the ordered state against quantum fluctuations. Effects of spin- and valley Zeeman fields as well as thermal fluctuations are also examined.

We study the coupling between a singlet-triplet qubit realized in a double quantum dot to a topological qubit realized by spatially well-separated Majorana bound states. We demonstrate that the singlet-triplet qubit can be leveraged for readout of the topological qubit and for supplementing the gate operations that cannot be performed by braiding of Majorana bound states. Furthermore, we extend our setup to a network of singlet-triplet and topological hybrid qubits that paves the way to scalable fault-tolerant quantum computing.

We study a Cooper pair transistor realized by a mesoscopic superconductor island that couples to a pair of $s$-wave superconducting leads. For a trivial island, the critical supercurrent between the leads exhibits a well-known $2e$-periodicity in the island-gate charge. Here, we show that for an island with spatially separated zero-energy Majorana or Andreev bound states the periodicity of the magnitude of the critical supercurrent transitions to $1e$ in the island-gate charge. Moreover, for Andreev bound states the current-phase relation displays a sign reversal when the parity of the charge ground state of the island changes between even and odd. Notably, for Majorana bound states the same sign reversal does not occur. Our results highlight the relevance of measuring the full current-phase relation of a Cooper pair transistor for clarifying the nature of zero-energy bound states in candidate systems for topological superconductors and provide an initial step towards integrating Majorana qubits in superconducting circuits.

We propose a universal gate set acting on a qubit formed by the degenerate ground states of a Coulomb-blockaded time-reversal invariant topological superconductor island with spatially separated Majorana Kramers pairs: the "Majorana Kramers Qubit". All gate operations are implemented by coupling the Majorana Kramers pairs to conventional superconducting leads. Interestingly, in such an all-superconducting device, the energy gap of the leads provides another layer of protection from quasiparticle poisoning independent of the island charging energy. Moreover, the absence of strong magnetic fields - which typically reduce the superconducting gap size of the island - suggests a unique robustness of our qubit to quasiparticle poisoning due to thermal excitations. Consequently, the Majorana Kramers Qubit should benefit from prolonged coherence times and may provide an alternative route to a Majorana-based quantum computer.

Adjoint functor theorems give necessary and sufficient conditions for a functor to admit an adjoint. In this paper we prove general adjoint functor theorems for functors between $\infty$-categories. One of our main results is an $\infty$-categorical generalization of Freyd's classical General Adjoint Functor Theorem. As an application of this result, we recover Lurie's adjoint functor theorems for presentable $\infty$-categories. We also discuss the comparison between adjunctions of $\infty$-categories and homotopy adjunctions, and give a treatment of Brown representability for $\infty$-categories based on Heller's purely categorical formulation of the classical Brown representability theorem., Comment: v1: 21 pages; v2: updated the references, minor changes; v3: 22 pages, changed the terminology from "final" to "coinitial" functors, added three further Corollaries 4.1.5, 5.1.4 and 5.1.5, additional minor changes, accepted for publication in the Journal of the London Mathematical Society

We propose a platform for universal quantum computation that uses conventional $s$-wave superconducting leads to address a topological qubit stored in spatially separated Majorana bound states in a multi-terminal topological superconductor island. Both the manipulation and read-out of this "Majorana superconducting qubit" are realized by tunnel couplings between Majorana bound states and the superconducting leads. The ability of turning on and off tunnel couplings on-demand by local gates enables individual qubit addressability while avoiding cross-talk errors. By combining the scalability of superconducting qubit and the robustness of topological qubits, the Majorana superconducting qubit may provide a promising and realistic route towards quantum computation.

We study a time-reversal-invariant topological superconductor island hosting spatially separated Majorana Kramers pairs, with weak tunnel couplings to two s-wave superconducting leads. When the topological superconductor island is in the Coulomb blockade regime, we predict that a Josephson current flows between the two leads due to a non-local transfer of Cooper pairs mediated by the Majorana Kramers pairs. Interestingly, we find that the sign of the Josephson current is controlled by the joint parity of all four Majorana bound states on the island. Consequently, this parity-controlled Josephson effect can be used for qubit read-out in Majorana-based quantum computing.

The thermal conductivity $\kappa$ of a material is an important parameter in many different applications. Optimization strategies of $\kappa$ often require insight into the dominant phonon scattering processes of the material under study. The Callaway model is widely used as an experimentalist's tool to analyze the lattice part of the thermal conductivity, $\kappa_l$. Here, we investigate how deviations from the implicitly assumed linear phonon dispersion relation affect $\kappa_l$ and in turn conclusions regarding the relevant phonon scattering processes. As an example, we show for the half-Heusler system (Hf,Zr,Ti)NiSn, that relying on the Callaway model in its simplest form has earlier resulted in a misinterpretation of experimental values by assigning the low measured $\kappa_l$ with unphysically strong phonon scattering in these materials. Instead, we propose an implementation of more realistic phonon dispersion curves, combined with empirical expressions for typical phonon scattering processes, which leads to far better quantitative agreement with both theoretical and experimental values. This method can easily be extended to other materials with known phonon dispersion relations.

We find a new 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.

Thermoelectric application of half-Heusler compounds suffers from their fairly high thermal conductivities. Insight into how effective various scattering mechanisms are in reducing the thermal conductivity of fabricated XNiSn compounds (X = Hf, Zr, Ti, and mixtures thereof) is therefore crucial. Here, we show that such insight can be obtained through a concerted theory-experiment comparison of how the lattice thermal conductivity kLat(T) depends on temperature and crystallite size. Comparing theory and experiment for a range of Hf0.5Zr0.5NiSn and ZrNiSn samples reported in the literature and in the present paper revealed that grain boundary scattering plays the most important role in bringing down kLat, in particular so for unmixed compounds. Our concerted analysis approach was corroborated by a good qualitative agreement between the measured and calculated kLat of polycrystalline samples, where the experimental average crystallite size was used as an input parameter for the calculations. The calculations were based on the Boltzmann transport equation and ab initio density functional theory. Our analysis explains the significant variation of reported kLat of nominally identical XNiSn samples and is expected to provide valuable insights into the dominant scattering mechanisms even for other materials.

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.

We consider the problem of sampling $n$ numbers from the range $\{1,\ldots,N\}$ without replacement on modern architectures. The main result is a simple divide-and-conquer scheme that makes sequential algorithms more cache efficient and leads to a parallel algorithm running in expected time $\mathcal{O}(n/p+\log p)$ on $p$ processors, i.e., scales to massively parallel machines even for moderate values of $n$. The amount of communication between the processors is very small (at most $\mathcal{O}(\log p)$) and independent of the sample size. We also discuss modifications needed for load balancing, online sampling, sampling with replacement, Bernoulli sampling, and vectorization on SIMD units or GPUs.

The interplay of superconductivity, magnetic fields, and spin-orbit interaction lies at the heart of topological superconductivity. Remarkably, the recent experimental discovery of $\varphi_{0}$ Josephson junctions by Szombati et al., Nat. Phys. 12, 568 (2016), characterized by a finite phase offset in the supercurrent, require the same ingredients as topological superconductors, which suggests a profound connection between these two distinct phenomena. Here, we theoretically show that a quantum dot $\varphi_{0}$ Josephson junction can serve as a new qualitative indicator for topological superconductivity: Microscopically, we find that the phase shift in a junction of $s-$wave superconductors is due to the spin-orbit induced mixing of singly occupied states on the qantum dot, while for a topological superconductor junction it is due to singlet-triplet mixing. Because of this important difference, when the spin-orbit vector of the quantum dot and the external Zeeman field are orthogonal, the $s$-wave superconductors form a $\pi$ Josephson junction while the topological superconductors have a finite offset $\varphi_{0}$ by which topological superconductivity can be distinguished from conventional superconductivity. Our prediction can be immediately tested in nanowire systems currently used for Majorana fermion experiments and thus offers a new and realistic approach for detecting topological bound states.

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 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 inter-qubit 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., Comment: 13 pages, 5 figures