Avenues of Majorana bound states (MBSs) have become one of the primary directions towards a possible realization of topological quantum computation. For a Y junction of Kitaev quantum wires, we numerically investigate the braiding of MBSs while considering the full quasiparticle background. The two central sources of braiding errors are found to be the fidelity loss due to the incomplete adiabaticity of the braiding operation as well as the finite hybridization of the MBSs. The explicit extraction of the braiding phase from the full many-particle states allows us to analyze the breakdown of the independent-particle picture of Majorana braiding. Furthermore, we find nearest-neighbor interactions to significantly affect the braiding performance for better or worse, depending on the sign and magnitude of the coupling.
Avenues of Majorana bound states (MBSs) have become one of the primary directions towards a possible realization of topological quantum computation. For a Y junction of Kitaev quantum wires, we numerically investigate the braiding of MBSs while considering the full quasiparticle background. The two central sources of braiding errors are found to be the fidelity loss due to the incomplete adiabaticity of the braiding operation as well as the finite hybridization of the MBSs. The explicit extraction of the braiding phase from the full many-particle states allows us to analyze the breakdown of the independent-particle picture of Majorana braiding. Furthermore, we find nearest-neighbor interactions to significantly affect the braiding performance for better or worse, depending on the sign and magnitude of the coupling.
Demonstration of a nontrivial result of quasiparticle exchange (or braiding) is usually considered the definitive proof of a topological phase with non-Abelian excitations, such as Majorana bound states (MBSs). However, in finite systems with disorder and smooth potential variations, the MBSs are imperfect in the sense that they are not fully isolated in space and can, to a varying degree, resemble conventional fermions. The main goal of this work is to reveal the braiding properties of isolated MBSs, regular fermions, and anything in between. We start from an experimentally attractive braiding protocol and find a way to compensate for the undesired splitting of the ground-state degeneracy which occurs during the protocol for imperfect MBSs. This leads to a braiding outcome that depends on the degree of MBS isolation but remains robust and non-Abelian except in the perfect fermion limit. Our protocol could be implemented in different platforms with non-Abelian excitations, including quantum-dot-based minimal Kitaev chains.
We numerically investigate non-Abelian braiding dynamics of vortices in two-dimensional topological superconductors, such as s-wave superconductors with Rashba spin-orbit coupling. Majorana zero modes (MZMs) hosted by the vortices constitute a topological qubit, which offers a fundamental building block of topological quantum computation. As the MZMs are protected by Z2 invariant, however, the Majorana qubit and quantum gate operations may be sensitive to intrinsic decoherence caused by MZM hybridization. Numerically simulating the time-dependent Bogoliubov--de Gennes equation without assuming a priori existence of MZMs, we examine quantum noises on the unitary operators of non-Abelian braiding dynamics due to interactions with neighboring MZMs and other quasiparticle states. We demonstrate that after the interchange of two vortices, the lowest vortex-bound states accumulate the geometric phase $\ensuremath{\pi}/2$, and errors stemming from dynamical phases are negligibly small, irrespective of interactions of MZMs. Furthermore, we numerically simulate the braiding dynamics of four vortices in two-dimensional topological superconductors, and discuss an optimal braiding condition for realizing the high performance of non-Abelian statistics and quantum gate operations of Majorana-based qubits.
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.
This thesis aims to investigate the novel physics around Majorana bound states (MBSs). MBSs are quasiparticle excitations predicted to exist in topological superconductors. Due to the inherent protection they obtain from their topology, they have long been regarded as a promising setup to build a topologically protected quantum computer. Apart from this very practical application, they continue to fascinate as platforms built with MBS exhibit unique effects like nonabelian braiding and non-Fermi liquid behaviour in the topological Kondo effect.The thesis starts with a general introduction to quantum computing, highlighting both the promises and challenges of that field. Afterwards follows an introduction to systems that are candidates to host MBSs. It focuses on the theoretical proposals and experimental realizations that are most relevant to the presented work. We continue with a summary of the most fundamental effects and the setups that make the physics of MBSs accessible. The author's contributions in this field are theoretical. However, each publication comes with the explicit goal of investigating setups that make the unique physics of MBSs directly accessible in experiments. The first three papers have been peer-reviewed and were published in scientific journals. The fourth one is currently under peer review. The first two papers investigate DC transport setups connecting to a qubit based on MBSs called the MBS tetron. We show that via an effect called parity blockade, it is possible to initialize the qubit, quantify its lifetime, and determine whether it holds MBSs or trivial (normal fermionic) states. Also employing parity blockade, the second paper establishes a transport setup that enables access to MBS fusion rules, an effect directly related to nonabelian braiding. The third paper translates the concept of a MBS tetron into the newly experimentally established approach of MBSs in quantum dot-based Kitaev chains. We denote this system the poor man's Majorana tetron. We find that the poor man's Majorana tetron in a transport setup with appropriate tuning exhibits the famed topological Kondo effect. Paper~IV focuses on nonabelian braiding. It investigates the hybridization-based braiding protocol and accounts for imperfections in MBS localization, breaking the ground state degeneracy. This detrimental effect is corrected by the introduction of a correction term, followed by a full analytical solution of the transformation induced by this generalized braiding protocol.
Majorana-based topological qubits are expected to exploit the nonabelian braiding statistics of Majorana modes in topological superconductors to realize fault-tolerant topological quantum computation.Scalable qubit designs require several Majorana modes localized on quantum wires networks, with braiding operations relying on the presence of the groundstate degeneracy of the topologically nontrivial superconducting phase.However, this degeneracy is lifted due to the hybridization between Majorana modes localized at a finite distance.Here, we describe a braiding protocol in a trijunction where each branch consists of a lattice of Majorana modes overlapping at a finite distance.We find that the energy splitting between the groundstate and the lowest-energy state decreases exponentially with the number of Majorana modes if the system is in its topologically nontrivial regime.This result does not rely on the specific braiding geometry and on the details of the braiding scheme but is a consequence of the supersymmetry and nontrivial topology of the effective low-energy Hamiltonian describing the Majorana lattice.
Majorana zero modes have emerged as one of the most promising platforms for topological quantum computation, since their non-Abelian braiding statistics allow quantum information to be encoded nonlocally and manipulated through braiding operations that are, in principle, protected against local perturbations. In practice, however, a braid is only as robust as its physical implementation: finite-time operation, residual couplings, and environmental noise can all convert local excitations into logical errors during the exchange process. Here, we address this question through a microscopic comparison of two representative braiding architectures, a nanowire trijunction and a quantum-dot-assisted setup, simulating the full time-dependent Bogoliubov--de Gennes dynamics under both noiseless and noisy conditions. We show that the dot-assisted architecture consistently achieves a lower error over a shorter timescale than the trijunction, owing to its more localized exchange mechanism. This advantage persists in the presence of noise, and a spatially resolved analysis further reveals that, in the dot-assisted geometry, fast noise localized on the dot produces a smaller error than equivalent noise on the wires, whereas slow, quasi-static noise on the dot becomes the dominant limitation. Taken together, these findings link the different error contributions directly to device geometry, pointing to concrete design principles for noise-resilient Majorana-based quantum gates.