Daniel Crawford

Majorana modes offer great potential for fault-tolerant quantum computation due to their topological protection. However, intrinsic disorder makes the unambiguous detection of Majorana modes difficult in the commonly used hybrid superconductor–semiconductor nanowire platform. By contrast, for magnet–superconductor hybrid systems, Majorana modes were theoretically predicted to be very robust against disorder, but no experimental confirmation of this has been reported so far. Here we demonstrate that these modes are indeed robust in one-dimensional spin chains constructed from individual iron atoms on a Rashba surface alloy with proximity-induced superconductivity. Although the chains exhibit perfect crystalline order, we observe nanoscale potential disorder of the BiAg ₂ /Ag(111)/Nb(110) heterostructure by scanning tunnelling microscopy. However, this does not prevent the emergence of zero-energy modes at both ends of the atomic chains, in agreement with tight-binding calculations showing that these modes are only found in the topologically non-trivial regime of the phase diagram. Our results explain the earlier observation of zero-energy Majorana modes in disordered Fe chains on other superconducting substrates, and may provide an avenue for the realization of Majorana qubits.

We show that vortices with circulating current, related with odd-frequency triplet pairing, appear in Josephson junctions where the barrier is a weak ferromagnet with strong spin-orbit coupling. By symmetry analysis we show that there is an additional term—a rotary invariant—in the superconducting free energy which allows for magnetoelectric effects even when the previously considered Lifshitz invariant vanishes. Using a microscopic model based on a modified Usadel equation incorporating those effects, we show that the size, shape, and position of these vortices can be controlled by manipulating Rashba spin-orbit coupling in the weak link, via gates, and we suggest that these vortices could be detected via scanning magnetometry techniques. We also show that the transverse triplet components of the pairing amplitudes can form a texture.

Qubits built out of Majorana zero modes (MZMs) have long been theorized as a potential pathway toward fault-tolerant topological quantum computation. Almost unavoidable in these processes is Majorana wave function overlap, known as hybridization, which arises throughout the process when Majorana modes get close to each other. This breaks the ground state degeneracy, leading to qubit errors in the braiding process. This work presents a simple but precise method to predict qubit errors for dynamic hybridization which varies in space and time. This includes hybridization between four or more MZMs through topological or trivial regions or both of them. As an illustration, we characterize qubit errors for an X gate. We demonstrate how to utilize the hybridization to implement not only arbitrary one-qubit rotations but also a two-qubit controlled variable phase gate, providing a demonstration of universal quantum computing.

Qubits built out of Majorana zero modes constitute the primary path toward topologically protected quantum computing. Simulating the braiding process of multiple Majorana zero modes corresponds to the quantum dynamics of a superconducting many-body system. It is crucial to study the Majorana dynamics both in the presence of all other quasiparticles and for reasonably large system sizes. We present a method to calculate arbitrary many-body wave functions as well as their expectation values, correlators, and overlaps from time evolved single-particle states of a superconductor, allowing for significantly larger system sizes. We calculate the fidelity, transition probabilities, and joint parities of Majorana pairs to track the quality of the braiding process. We show how the braiding success depends on the speed of the braid. Moreover, we demonstrate the topological CNOT two-qubit gate as an example of two-qubit entanglement. Our Letter opens the path to test and analyze the many theoretical implementations of Majorana qubits. Moreover, this method can be used to study the dynamics of any noninteracting superconductor.

Magnet-superconductor hybrid (MSH) systems are a key platform for custom-designed topological superconductors. Ideally, the ends of a one-dimensional MSH structure will host Majorana zero-modes (MZMs), the fundamental unit of topological quantum computing. However, some experiments with ferromagnetic (FM) chains show a more complicated picture. Due to tiny gap sizes and hence long coherence lengths, MZMs might hybridize and lose their topological protection. Recent experiments on a niobium surface have shown that both FM and antiferromagnetic (AFM) chains may be engineered, with the magnetic order depending on the crystallographic direction of the chain. While FM chains are well understood, AFM chains are less so. Here, we study two models inspired by the niobium surface: A minimal model to elucidate the general topological properties of AFM chains and an extended model to more closely simulate a real system by mimicking the proximity effect. We find that, in general, for AFM chains, the topological gap is larger than for FM ones, and thus, coherence lengths are shorter for AFM chains, yielding more pronounced localization of MZMs in these chains. While for some parameters AFM chains may be topologically trivial, we find in these cases that adding an adjacent chain can result in a nontrivial system, with a single MZM at each chain end.

Magnet-superconductor hybrid (MSH) systems represent promising platforms to host Majorana zero modes (MZMs), the elemental building blocks for fault-tolerant quantum computers. Theoretical description of such MSH structures is mostly based on simplified models, not accounting for the complexity of real materials. Here, based on density functional theory, we derive a superconducting 80-band model to study an MSH system consisting of a magnetic manganese chain on the s wave superconductor niobium. For a wide range of values of the superconducting order parameter, the system is a topological superconductor, with MZMs exhibiting non-universal spatial patterns and a drastic accumulation of spectral weight on both sides along the magnetic chain. These side feature states can be explained by an effective model which is guided by the ab initio results. Performing scanning tunneling spectroscopy experiments on the same system, we observe a spatial structure in the low-energy local density of states that is consistent with the theoretical findings. Our results open a first-principle approach to the discovery of topological superconductors.