# Parity protected quantum computing in superconductors

- Paritätsgeschütztes Quantum Computing in Supraleitern

Otten, Daniel; Hassler, Fabian (Thesis advisor); DiVincenzo, David (Thesis advisor)

*Aachen (2019)*

Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2019

Abstract

A fully functional quantum computer as a device that is capable of enhancing certain important classical computer algorithms, like the factoring of prime numbers, is still a dream for the future. The major problem for the engineering of a quantum computer is the fragility of quantum systems due to interactions with their environments. These effects make the qubits, which are the quantum equivalents to the classical bits, hard to control and hard to maintain in a stable state. The idea of using topologically protected quantum systems as the physical basis of qubits is an approach to handle the problem of occurring errors by directly protecting the computational subspace. In this thesis we treat two such concepts that both are based on parity protection. The parity is a quantity that differentiates between even and odd values of a variable. More explicitly, we treat the scheme of topological protection due to Majorana zero modes that is physically based on the protection of the fermion parity in a quantum system. Furthermore, we consider the scheme of the $0$-$\pi$ qubit in superconducting circuits that is based on the protection of the parity of the superconducting phase, discriminating between phases of even and odd multiples of $\pi$. In the first part of this thesis, we review the two schemes and point out similarities between them. We argue that these similarities origin from the fact that the schemes are dual to each other. In the second part of this thesis, we set the focus on calculating physical quantities that can be used to experimentally detect new routes of improving the two presented schemes of parity protected quantum computing. First, we introduce the concept of quantum phase slips and consider a chain of strongly coupled transmon qubits to show how these phase slips can mediate interactions along the chain. The correct handling of such interactions is relevant in the process of qubit engineering like the $0$-$\pi$ qubit. Moreover, we propose a new parameter regime for the measurement of the so-called Korshunov instantons in superconducting circuits. The existence of these instantons implies that the system can be described by a tunnel junction subject to dissipative quasiparticle tunneling. The action corresponding to such a tunnel junction is periodic in the superconducting phase and turns out to support coherent bands not influenced by the dissipative dynamics. We suggest to use this dissipative effect to establish a further protection for the $0$-$\pi$ qubit. In the last part of the thesis, we analyze Kitaev's honeycomb model describing spins on a lattice. The spin degrees of freedom fractionalize into Majorana degrees moving on the lattice in the background of a static $\mathbb{Z}_2$ gauge field. For certain choices of parameters, the model supports Majorana zero modes. We calculate the wave functions of these zero modes and the energy splitting between two hybridized modes. Furthermore, we calculate the dynamical structure factor in the presence of a quenched disorder configuration of the zero modes. Especially the results for the latter can help in the experimental characterization of materials that host Majorana zero modes and thus are possibly useful for parity protected quantum computing.

Institutions

- Department of Physics [130000]
- Theoretical Physics (condensed matter) Teaching and Research Area [135220]

### Identifier

- DOI: 10.18154/RWTH-2019-05500
- RWTH PUBLICATIONS: RWTH-2019-05500