Michael Wimmer

Quantum dot (QD)-superconductor hybrids have been established as a suitable platform for realizing Kitaev chains hosting Majorana bound states. Implementing these structures in a qubit architecture is expected to result in coherence times that scale exponentially with the lengths of the chains. To scale to longer systems, the phase differences between all superconducting segments in the chain need to be controlled. While this control has been demonstrated by using an external magnetic flux [ten Haaf , ; Bordin , ], ideally, it can be achieved with control over intrinsic system parameters [Liu , ]. In this work, we investigate whether the relevant phase differences can be tuned through the spin degree of freedom in each QD or the chemical potential of the discrete bound states in the hybrid sections. We confirm that both these tuning knobs allow for controlling the phase difference in the couplings between neighboring QDs, bypassing the requirement to tune an external flux. However, we find that the amplitude of the phase shifts can deviate from a discrete π shift. We introduce a spatial variation in the spin-orbit field as a possible mechanism to explain the observed behavior and comment on the consequences for experimentally creating long Kitaev chains.

The Fermi-Hubbard model describes a large variety of condensed matter systems with spinful fermions and strong interactions. On the other hand, the Kitaev chain model deals with noninteracting spinless fermions and produces robust ground-state degeneracies that give rise to Majorana bound states. In this Letter, we connect these two domains by experimentally studying small arrays of quantum dots (QDs) coupled via superconductors. Without a magnetic field, this system constitutes a Fermi-Hubbard model with intersite superconducting correlations. Using two coupled QDs, we find sweet spots in parameter space where we observe zero-bias conductance peaks that are robust against on-site perturbations of the QDs, similar to the behavior expected of a two-site Kitaev chain. This observation is consistent with the presence of Majorana Kramers pairs or Z 3 parafermions in this minimal system. Extending to three sites, we find that the spinful system scales very differently compared to the spinless Kitaev chain. When the same sweet-spot conditions are satisfied, the ground state degeneracy of the full three-site system is lifted. The degeneracy can be restored by tuning the superconducting phase difference between the hybrid segments, but the robustness against on-site detunings is lost. Our findings are a first step in studying ground-state degeneracies in strongly interacting Fermi-Hubbard systems with superconducting correlations.

Protecting qubits from perturbations is a central challenge in quantum computing. Topological superconductors with separated Majorana bound states (MBSs) provide a strong form of protection that only depends on the locality of perturbations. While the link between MBS separation, robust degeneracy, and protected braiding is well understood in noninteracting systems, recent experimental progress in short quantum-dot-based Kitaev chains highlights the need to establish these connections rigorously for interacting systems. We do this by defining MBSs from many-body ground states and show how their locality constrains their coupling to an environment. This, in turn, quantifies the protection of the energy degeneracy and the feasibility of non-abelian braiding.

To observe Majorana bound states, and especially to use them as a qubit, requires careful optimization of competing quality metrics. We systematically compare Majorana quality in proximitized semiconductor nanowires and quantum dot chains. Using multi-objective optimization, we analyze the fundamental trade-offs between topological gap and localization length, two key metrics that determine MBS coherence and operational fidelity. We demonstrate that these quantities cannot be simultaneously optimized in realistic models, creating Pareto frontiers that define the achievable parameter space. Our results show that QD chains achieve both comparable quality as nanowires and a regime with a much shorter localization length, making them particularly promising for near-term quantum computing applications where device length and disorder are limiting factors.

The easily accessible experimental signatures of Majorana modes are ambiguous and only probe topology indirectly: for example, quasi-Majorana states mimic most properties of Majoranas. Establishing a correspondence between an experiment and a theoretical model known to be topological resolves this ambiguity. Here we demonstrate that already theoretically determining whether a finite system is topological is by itself ambiguous. In particular, we show that the scattering topological invariant—a probe of topology most closely related to transport signatures of Majoranas—has multiple biases in finite systems. For example, we identify that quasi-Majorana states also mimic the scattering invariant of Majorana zero modes in intermediate-sized systems. We expect that the bias due to finite size effects is universal, and advocate that the analysis of topology in finite systems should be accompanied by a comparison with the thermodynamic limit. Our results are directly relevant to the applications of the topological gap protocol.

We theoretically explore the emergence of strong zero modes in a two-site chain consisting of two quantum dots coupled due to a central dot that mediates electron hopping and singlet superconducting pairing. In the presence of time-reversal symmetry, the on-site Coulomb interaction leads to a three-fold ground-state degeneracy when tuning the system to a sweet spot as a function of the inter-dot couplings. This degeneracy is protected against changes of the dot energies in the same way as “poor man’s” Majorana bound states in short Kitaev chains. In the limit of strong interactions, this protection is maximal and the entire spectrum becomes triply degenerate, indicating the emergence of a “poor man’s” version of a strong zero mode. We explain the degeneracy and protection by constructing corresponding Majorana Kramers-pair operators and \mathbb{Z}_3 ℤ 3 -parafermion operators. The strong zero modes share many properties of Majorana bound states in short Kitaev chains, including the stability of zero-bias peaks in the conductance and the behavior upon coupling to an additional quantum dot. However, they can be distinguished through finite-bias spectroscopy and the exhibit a different behavior when scaling to longer chains.

A chain of quantum dots (QDs) in semiconductor–superconductor hybrid systems can form an artificial Kitaev chain hosting Majorana bound states (MBSs)^1,2,3. These zero-energy states are expected to be localized on the edges of the chain⁴, at the outermost QDs. The remaining QDs, comprising the bulk, are predicted to host an excitation gap that protects the MBSs at the edges from local on-site perturbations. Here we demonstrate this connection between the bulk and edges in a minimal system, by engineering a three-site Kitaev chain in a two-dimensional electron gas. Through direct tunnelling spectroscopy on each site, we show that the appearance of stable zero-bias conductance peaks at the outer QDs is correlated with the presence of an excitation gap in the middle QD. Furthermore, we show that this gap can be controlled by applying a superconducting phase difference between the two hybrid segments and that the MBSs are robust only when the excitation gap is present. We find a close agreement between experiments and the original Kitaev model, thus confirming key predictions for MBSs in a three-site chain.

Majorana zero modes are non-Abelian quasiparticles predicted to emerge at the edges of topological superconductors. A one-dimensional topological superconductor can be realized with the Kitaev model—a chain of spinless fermions coupled via p-wave superconductivity and electron hopping—which becomes topological in the long-chain limit. Here we realize a three-site Kitaev chain using semiconducting quantum dots coupled by superconducting segments in a hybrid InSb/Al nanowire. We investigate the robustness of Majorana zero modes under varying coupling strengths and electrochemical potentials, comparing two- and three-site chains realized within the same device. We observe that extending the chain to three sites enhances the stability of the zero-energy modes, especially against variations in the coupling strengths. This experiment lacks superconducting phase control, yet numerical conductance simulations with phase averaging align well with our observations. Our results demonstrate the scalability of quantum-dot-based Kitaev chains and its benefits for Majorana stability.

Quantum-dot-superconductor arrays have emerged as a new and promising material platform for realizing topological Kitaev chains. So far, experiments have implemented a two-site chain with limited protection. Here, we propose an experimentally feasible protocol for scaling up the chain in order to enhance the protection of the Majorana zero modes. To this end, we make use of the fact that the relative sign of normal and superconducting hoppings mediated by an Andreev bound state can be changed by electrostatic gates. In this way, our method only relies on the use of individual electrostatic gates on hybrid regions, quantum dots, and tunnel barriers, respectively, without the need for individual magnetic flux control, greatly simplifying the device design. Our work provides guidance for realizing a topologically protected Kitaev chain, which is the building block of error-resilient topological quantum computation. Published by the American Physical Society 2025

We demonstrate that the classical dynamics influence the localization behaviour of Majorana wavefunctions in Majorana billiards. By using a connection between Majorana wavefunctions and eigenfunctions of a normal state Hamiltonian, we show that Majorana wavefunctions in both p-wave and s-wave topological superconductors inherit the properties of the underlying normal state eigenfunctions. As an example, we demonstrate that Majorana wavefunctions in topological superconductors with chaotic shapes feature quantum scarring. Furthermore, we show a way to manipulate a localized Majorana wavefunction by altering the underlying classical dynamics using a local potential away from the localization region. Finally, in the presence of chiral symmetry breaking, we find that the Majorana wavefunction in convex-shaped Majorana billiards exhibits caustics formation, reminiscent of a normal state system with magnetic field.