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The fusion of non-Abelian anyons or topological defects is a fundamental operation in measurement-only topological quantum computation. In topological superconductors, this operation amounts to a determination of the shared fermion parity of Majorana zero modes. As a step towards this, we implement a single-shot interferometric measurement of fermion parity in indium arsenide-aluminum heterostructures with a gate-defined nanowire. The interferometer is formed by tunnel-coupling the proximitized nanowire to quantum dots. The nanowire causes a state-dependent shift of these quantum dots' quantum capacitance of up to 1 fF. Our quantum capacitance measurements show flux h/2e-periodic bimodality with a signal-to-noise ratio of 1 in 3.7 $\mu$s at optimal flux values. From the time traces of the quantum capacitance measurements, we extract a dwell time in the two associated states that is longer than 1 ms at in-plane magnetic fields of approximately 2 T. These results are consistent with a measurement of the fermion parity encoded in a pair of Majorana zero modes that are separated by approximately 3 $\mu$m and subjected to a low rate of poisoning by non-equilibrium quasiparticles. The large capacitance shift and long poisoning time enable a parity measurement error probability of 1%., Comment: Added data on a second measurement of device A and a measurement of device B, expanded discussion of a trivial scenario. Refs added, author list updated

We present measurements and simulations of semiconductor-superconductor heterostructure devices that are consistent with the observation of topological superconductivity and Majorana zero modes. The devices are fabricated from high-mobility two-dimensional electron gases in which quasi-one-dimensional wires are defined by electrostatic gates. These devices enable measurements of local and non-local transport properties and have been optimized via extensive simulations to ensure robustness against non-uniformity and disorder. Our main result is that several devices, fabricated according to the design's engineering specifications, have passed the topological gap protocol defined in Pikulin et al. [arXiv:2103.12217]. This protocol is a stringent test composed of a sequence of three-terminal local and non-local transport measurements performed while varying the magnetic field, semiconductor electron density, and junction transparencies. Passing the protocol indicates a high probability of detection of a topological phase hosting Majorana zero modes as determined by large-scale disorder simulations. Our experimental results are consistent with a quantum phase transition into a topological superconducting phase that extends over several hundred millitesla in magnetic field and several millivolts in gate voltage, corresponding to approximately one hundred micro-electron-volts in Zeeman energy and chemical potential in the semiconducting wire. These regions feature a closing and re-opening of the bulk gap, with simultaneous zero-bias conductance peaks at both ends of the devices that withstand changes in the junction transparencies. The extracted maximum topological gaps in our devices are 20-60 $\mu$eV. This demonstration is a prerequisite for experiments involving fusion and braiding of Majorana zero modes., Comment: Final version

The coherent superposition of non-orthogonal fermionic Gaussian states has been shown to be an efficient approximation to the ground states of quantum impurity problems [Bravyi and Gosset,Comm. Math. Phys.,356 451 (2017)]. We present a practical approach for performing a variational calculation based on such states. Our method is based on approximate imaginary-time equations of motion that decouple the dynamics of each Gaussian state forming the ansatz. It is independent of the lattice connectivity of the model and the implementation is highly parallelizable. To benchmark our variational method, we calculate the spin-spin correlation function and R\'enyi entanglement entropy of an Anderson impurity, allowing us to identify the screening cloud and compare to density matrix renormalization group calculations. Secondly, we study the screening cloud of the two-channel Kondo model, a problem difficult to tackle using existing numerical tools., Comment: 16 pages, 7 figures

Amplifiers based on Josephson junctions allow for a fast and noninvasive readout of superconducting qubits. Motivated by the ongoing progress toward the realization of fault-tolerant qubits based on Majorana bound states, we investigate the topological counterpart of the Josephson bifurcation amplifier. We predict that the bifurcation dynamics of a topological Josephson junction driven in the appropriate parameter regime may be used as an additional tool to detect the emergence of Majorana bound states., Comment: 16 pages, 11 figures

Solid-state experimental realizations of Majorana bound states are based on materials with strong intrinsic spin-orbit interactions. In this paper, we explore an alternative approach where spin-orbit coupling is induced artificially through a nonuniform magnetic field that originates from an array of micromagnets. Using a recently developed optimization algorithm, we find suitable magnet geometries for the emergence of topological superconductivity in wires without intrinsic spin-orbit coupling. We confirm the robustness of Majorana bound states against disorder and periodic potentials whose amplitudes do not exceed the Zeeman energy. Furthermore, we identify low g-factor materials commonly used in mesoscopic physics experiments as viable candidates for Majorana devices., Comment: 13 pages, 9 figures, published version

We present a many-body exact diagonalization study of the $\mathbb{Z}_2$ and $\mathbb{Z}_4$ Josephson effects in circuit quantum electrodynamics architectures. Numerical simulations are conducted on Kitaev chain Josephson junctions hosting nearest-neighbor Coulomb interactions. The low-energy effective theory of highly transparent Kitaev chain junctions is shown to be identical to that of junctions created at the edge of a quantum spin-Hall insulator. By capacitively coupling the interacting junction to a microwave resonator, we predict signatures of the fractional Josephson effects on the cavity frequency and on time-resolved reflectivity measurements., Comment: 16 pages, 11 figures

Recent progress toward the fabrication of Majorana-based qubits has sparked the need for systematic approaches to optimize experimentally relevant parameters for the realization of robust Majorana bound states. Here, we introduce an efficient numerical method for the real-space optimization of tunable parameters, such as electrostatic potential profiles and magnetic field textures, in Majorana wires. Combining ideas from quantum control and quantum transport, our algorithm, applicable to any noninteracting tight-binding model, operates on a largely unexplored parameter space and opens new routes for Majorana bound states with enhanced robustness. Contrary to common belief, we find that spatial inhomogeneities of parameters can be a resource for the engineering of Majorana bound states., Comment: 15 pages including appendices, 11 figures

Recent theoretical work has established the presence of hidden spin and orbital textures in non-magnetic materials with inversion symmetry. Here, we propose that these textures can be detected by nuclear magnetic resonance (NMR) measurements carried out in the presence of an electric field. In crystals with hidden polarizations, a uniform electric field produces a staggered magnetic field that points to opposite directions at atomic sites related by spatial inversion. As a result, the NMR resonance peak corresponding to inversion partner nuclei is split into two peaks. The magnitude of the splitting is proportional to the electric field and depends on the orientation of the electric field with respect to the crystallographic axes and the external magnetic field. As a case study, we present a theory of electric-field-induced splitting of NMR peaks for $^{77}$Se, $^{125}$Te and $^{209}$Bi in Bi$_2$Se$_3$ and Bi$_2$Te$_3$. In conducting samples with current densities of $\simeq 10^6\, {\rm A/cm}^2$, the splitting for Bi can reach $100\, {\rm kHz}$, which is comparable to or larger than the intrinsic width of the NMR lines. In order to observe the effect experimentally, the peak splitting must also exceed the linewidth produced by the Oersted field. In Bi$_2$Se$_3$, this requires narrow wires of radius $\lesssim 1\, \mu{\rm m}$. We also discuss other potentially more promising candidate materials, such as SrRuO$_3$ and BaIr$_2$Ge$_2$, whose crystal symmetry enables strategies to suppress the linewidth produced by the Oersted field., Comment: 13 pages, 8 figures

Single-mode Josephson junction-based parametric amplifiers are often modeled as perfect amplifiers and squeezers. We show that, in practice, the gain, quantum efficiency, and output field squeezing of these devices are limited by usually neglected higher-order corrections to the idealized model. To arrive at this result, we derive the leading corrections to the lumped-element Josephson parametric amplifier of three common pumping schemes: monochromatic current pump, bichromatic current pump, and monochromatic flux pump. We show that the leading correction for the last two schemes is a single Kerr-type quartic term, while the first scheme contains additional cubic terms. In all cases, we find that the corrections are detrimental to squeezing. In addition, we show that the Kerr correction leads to a strongly phase-dependent reduction of the quantum efficiency of a phase-sensitive measurement. Finally, we quantify the departure from ideal Gaussian character of the filtered output field from numerical calculation of third and fourth order cumulants. Our results show that, while a Gaussian output field is expected for an ideal Josephson parametric amplifier, higher-order corrections lead to non-Gaussian effects which increase with both gain and nonlinearity strength. This theoretical study is complemented by experimental characterization of the output field of a flux-driven Josephson parametric amplifier. In addition to a measurement of the squeezing level of the filtered output field, the Husimi Q-function of the output field is imaged by the use of a deconvolution technique and compared to numerical results. This work establishes nonlinear corrections to the standard degenerate parametric amplifier model as an important contribution to Josephson parametric amplifier's squeezing and noise performance., Comment: 21 pages, 13 figures

We study an implementation of the open GRAPE (Gradient Ascent Pulse Engineering) algorithm well suited for large open quantum systems. While typical implementations of optimal control algorithms for open quantum systems rely on explicit matrix exponential calculations, our implementation avoids these operations leading to a polynomial speed-up of the open GRAPE algorithm in cases of interest. This speed-up, as well as the reduced memory requirements of our implementation, are illustrated by comparison to a standard implementation of open GRAPE. As a practical example, we apply this open-system optimization method to active reset of a readout resonator in circuit QED. In this problem, the shape of a microwave pulse is optimized such as to empty the cavity from measurement photons as fast as possible. Using our open GRAPE implementation, we obtain pulse shapes leading to a reset time over four times faster than passive reset., Comment: 11 pages, 6 figures

Cat states of the microwave field stored in high-Q resonators show great promise for robust encoding and manipulation of quantum information. Here we propose an approach to efficiently prepare such cat states in a Kerr-nonlinear resonator by the use of a two-photon drive. We show that this preparation is robust against single-photon loss. We moreover find that it is possible to remove undesirable phase evolution induced by a Kerr nonlinearity using a two-photon drive of appropriate amplitude and phase. Finally, we present a universal set of quantum logical gates that can be performed on the engineered eigenspace of the two-photon driven Kerr-nonlinear resonator.

Motivated by recent nuclear magnetic resonance (NMR) experiments, we present a microscopic sp3 tight-binding model calculation of the NMR shifts in bulk Bi2Se3, and Bi2Te3. We compute the contact, dipolar, orbital and core polarization contributions to the carrier-density-dependent part of the NMR shifts in Bi209, Te125 and Se77. The spin-orbit coupling and the layered crystal structure result in a contact Knight shift with strong uniaxial anisotropy. Likewise, because of spin-orbit coupling, dipolar interactions make a significant contribution to the isotropic part of the NMR shift. The contact interaction dominates the isotropic Knight shift in Bi209 NMR, even though the electronic states at the Fermi level have a rather weak s-orbital character. In contrast, the contribution from the contact hyperfine interaction to the NMR shift of Se77 and Te125 is weak compared to the dipolar and orbital shifts therein. In all cases, the orbital shift is at least comparable to the contact and dipolar shifts, while the shift due to core polarization is subdominant (except for Te nuclei located at the inversion centers). By artificially varying the strength of spin-orbit coupling, we evaluate the evolution of the NMR shift across a band inversion but find no clear signature of the topological transition., Comment: 21 pages, 8 figures