Katharina Laubscher


Department of Physics
University of Basel
Klingelbergstrasse 82
CH-4056 Basel, Switzerland

email:view address

tel: +41 61 207 36 95

Short CV

2018 - present:PhD student in the Condensed Matter Theory & Quantum Computing Group at the University of Basel, supervised by Prof. Dr. Jelena Klinovaja and Prof. Dr. Daniel Loss (QCQT Fellowship)
2014 - 2017:Master of Science in Physics, University of Basel
Master's thesis: Universal quantum computation using a hybrid quantum double model, supervised by Dr. James Wootton and Prof. Dr. Daniel Loss
2011 - 2014:Bachelor of Science in Physics, University of Basel

Research interests

Majorana fermions and parafermions in condensed matter systems, topological quantum computation


Show all abstracts.

1.  Majorana bound states in semiconducting nanostructures
Katharina Laubscher and Jelena Klinovaja.

In this Tutorial, we give a pedagogical introduction to Majorana bound states (MBSs) arising in semiconducting nanostructures. We start by briefly reviewing the well-known Kitaev chain toy model in order to introduce some of the basic properties of MBSs before proceeding to describe more experimentally relevant platforms. Here, our focus lies on simple `minimal' models where the Majorana wave functions can be obtained explicitly by standard methods. In a first part, we review the paradigmatic model of a Rashba nanowire with strong spin-orbit interaction (SOI) placed in a magnetic field and proximitized by a conventional s-wave superconductor. We identify the topological phase transition separating the trivial phase from the topological phase and demonstrate how the explicit Majorana wave functions can be obtained in the limit of strong SOI. In a second part, we discuss MBSs engineered from proximitized edge states of two-dimensional (2D) topological insulators. We introduce the Jackiw-Rebbi mechanism leading to the emergence of bound states at mass domain walls and show how this mechanism can be exploited to construct MBSs. Due to their recent interest, we also include a discussion of Majorana corner states in 2D second-order topological superconductors. This Tutorial is mainly aimed at graduate students -- both theorists and experimentalists -- seeking to familiarize themselves with some of the basic concepts in the field.

2.  Fractional boundary charges with quantized slopes in interacting one- and two-dimensional systems
Katharina Laubscher, Clara S. Weber, Dante M. Kennes, Mikhail Pletyukhov, Herbert Schoeller, Daniel Loss, and Jelena Klinovaja.

We study fractional boundary charges (FBCs) for two classes of strongly interacting systems. First, we study strongly interacting nanowires subjected to a periodic potential with a period that is a rational fraction of the Fermi wavelength. For sufficiently strong interactions, the periodic potential leads to the opening of a charge density wave gap at the Fermi level. The FBC then depends linearly on the phase offset of the potential with a quantized slope determined by the period. Furthermore, different possible values for the FBC at a fixed phase offset label different degenerate ground states of the system that cannot be connected adiabatically. Next, we turn to the fractional quantum Hall effect (FQHE) at odd filling factors ν=1/(2l+1), where l is an integer. For a Corbino disk threaded by an external flux, we find that the FBC depends linearly on the flux with a quantized slope that is determined by the filling factor. Again, the FBC has 2l+1 different branches that cannot be connected adiabatically, reflecting the (2l+1)-fold degeneracy of the ground state. These results allow for several promising and strikingly simple ways to probe strongly interacting phases via boundary charge measurements.

3.  Kramers pairs of Majorana corner states in a topological insulator bilayer
Katharina Laubscher, Danial Chughtai, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 102, 195401 (2020); arXiv:2007.13579.

We consider a system consisting of two tunnel-coupled two-dimensional topological insulators proximitized by a top and bottom superconductor with a phase difference of π between them. We show that this system exhibits a time-reversal invariant second-order topological superconducting phase characterized by the presence of a Kramers pair of Majorana corner states at all four corners of a rectangular sample. We furthermore investigate the effect of a weak time-reversal symmetry breaking perturbation and show that an in-plane Zeeman field leads to an even richer phase diagram exhibiting two nonequivalent phases with two Majorana corner states per corner as well as an intermediate phase with only one Majorana corner state per corner. We derive our results analytically from continuum models describing our system. In addition, we also provide independent numerical confirmation of the resulting phases using discretized lattice representations of the models, which allows us to demonstrate the robustness of the topological phases and the Majorana corner states against parameter variations and potential disorder.

4.  Majorana zero modes and their bosonization
Victor Chua, Katharina Laubscher, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 102, 155416 (2020); arXiv:2006.03344.

The simplest continuum model of a one-dimensional non-interacting superconducting fermionic symmetry-protected topological (SPT) phase is analyzed in great detail using analytic methods. A full exact diagonalization of the mean-field Bogoliubov-de Gennes Hamiltonian is carried out with open boundaries and finite lengths. Majorana zero modes are derived and studied in great detail. Thereafter exact operator bosonization in both open and closed geometries is carried out. The complementary viewpoints provided by fermionic and bosonic formulations of the superconducting SPT phase are then reconciled. In particular, we provide a complete and exact account of how the topological Majorana zero modes manifest in a bosonized description of an SPT phase.

5.  Majorana and parafermion corner states from two coupled sheets of bilayer graphene
Katharina Laubscher, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. Research 2, 013330 (2020); arXiv:1912.10931.

We consider a setup consisting of two coupled sheets of bilayer graphene in the regime of strong spin-orbit interaction, where electrostatic confinement is used to create an array of effective quantum wires. We show that for suitable interwire couplings the system supports a topological insulator phase exhibiting Kramers partners of gapless helical edge states, while the additional presence of a small in-plane magnetic field and weak proximity-induced superconductivity leads to the emergence of zero-energy Majorana corner states at all four corners of a rectangular sample, indicating the transition to a second-order topological superconducting phase. The presence of strong electron-electron interactions is shown to promote the above phases to their exotic fractional counterparts. In particular, we find that the system supports a fractional topological insulator phase exhibiting fractionally charged gapless edge states and a fractional second-order topological superconducting phase exhibiting zero-energy Z_{2m} parafermion corner states, where m is an odd integer determined by the position of the chemical potential.

6.  Fractional topological superconductivity and parafermion corner states
Katharina Laubscher, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. Research 1, 032017(R) (2019); arXiv:1905.00885.

We consider a system of weakly coupled Rashba nanowires in the strong spin-orbit interaction (SOI) regime. The nanowires are arranged into two tunnel-coupled layers proximitized by a top and bottom superconductor such that the superconducting phase difference between them is π. We show that in such a system strong electron-electron interactions can stabilize a helical topological superconducting phase hosting Kramers partners of ℤ2m parafermion edge modes, where m is an odd integer determined by the position of the chemical potential. Furthermore, upon turning on a weak in-plane magnetic field, the system is driven into a second-order topological superconducting phase hosting zero-energy ℤ2m parafermion bound states localized at two opposite corners of a rectangular sample. As a special case, zero-energy Majorana corner states emerge in the non-interacting limit m=1, where the chemical potential is tuned to the SOI energy of the single nanowires.

7.  Universal quantum computation in the surface code using non-Abelian islands
Katharina Laubscher, Daniel Loss, and James R. Wootton.
Phys. Rev. A 100, 012338 (2019); arxiv:1811.06738.

The surface code is currently the primary proposed method for performing quantum error correction. However, despite its many advantages, it has no native method to fault-tolerantly apply non-Clifford gates. Additional techniques are therefore required to achieve universal quantum computation. Here we propose a new method, using small islands of a qudit variant of the surface code. This allows the non-trivial action of the non-Abelian anyons in the latter to process information stored in the former. Specifically, we show that a non-stabilizer state can be prepared, which allows universality to be achieved.

8.  Poking holes and cutting corners to achieve Clifford gates with the surface code
Benjamin J. Brown, Katharina Laubscher, Markus S. Kesselring, and James R. Wootton.
Phys. Rev. X 7, 021029 (2017); arXiv:1609.04673.

The surface code is currently the leading proposal to achieve fault-tolerant quantum computation. Among its strengths are the plethora of known ways in which fault-tolerant Clifford operations can be performed, namely, by deforming the topology of the surface, by the fusion and splitting of codes and even by braiding engineered Majorana modes using twist defects. Here we present a unified framework to describe these methods, which can be used to better compare different schemes, and to facilitate the design of hybrid schemes. Our unification includes the identification of twist defects with the corners of the planar code. This identification enables us to perform single-qubit Clifford gates by exchanging the corners of the planar code via code deformation. We analyse ways in which different schemes can be combined, and propose a new logical encoding. We also show how all of the Clifford gates can be implemented with the planar code without loss of distance using code deformations, thus offering an attractive alternative to ancilla-mediated schemes to complete the Clifford group with lattice surgery.