Abstract:  
In this thesis, we study the topological classification of symmetric quantum walks. These describe the discrete time evolution of single quantum particles on the lattice with additional locally acting symmetries. The thesis consists of three parts:
In the first part, we discuss discrete symmetry types for selfadjoint and unitary operators from an abstract point of view, i.e. without assuming an underlying physical model. We reduce any abstract finite group of involutive symmetries and their projective representations to a smaller set of symmetry types, eliminating elements that are redundant for topological classifications. This reduction process leads to the wellknown tenfold way for selfadjoint operators, and for unitary operators, we identify 38 nonredundant symmetry types. For these, we define a symmetry index, which labels equivalence classes of finitedimensional representations up to trivial direct summands. We show that these equivalence classes naturally carry a group structure and finish the discussion by explicitly computing the corresponding index groups for all nontrivial symmetry types.
Second, we develop a topological classification for symmetric quantum walks based on the symmetry index derived in the first part. We begin without a locality condition on the unitary time evolution operator but only assume an underlying discrete spatial structure. Unlike continuoustime systems, quantum walks exhibit nongentle perturbations, i.e. local or compact perturbations that cannot be undone continuously. Using the symmetry index, we provide a complete topological classification of such perturbations of unitary operators on any lattice or graph. We add a locality condition on the onedimensional lattice and detail the implications of such assumption on the classification. The spatial structure of the onedimensional lattice allows us to define the left and right symmetry index, which characterise a walks topological properties on the two halfchains.
The sum of these two indices equals the overall symmetry index, which provides a lower bound on the number of symmetry protected eigenstates of the walk. For the symmetry types of the tenfold way, a subset of three different symmetry indices is complete with respect to normcontinuous deformations and compact perturbations.
In the third part, we consider quantum walk protocols instead of single timestep unitaries. We show that any unitary operator with finite jump length on a onedimensional lattice can be factorised into a sequence of shift and coin operations. We then provide a complete topological classification of such protocols under the influence of chiral symmetry. The classification is in terms of the halfstep operator, i.e. the time evolution operator at half of the driving period, which is singled out by the chiral symmetry. We also show that a halfstep operator can be constructed for every chiral symmetric single timestep unitary without a predefined underlying protocol. This renders the classification via the halfstep operator valid for periodically driven continuoustime (Floquet systems), discretely driven protocols, and single timestep quantum walks.


License of this version:  Es gilt deutsches Urheberrecht. Das Dokument darf zum eigenen Gebrauch kostenfrei genutzt, aber nicht im Internet bereitgestellt oder an Außenstehende weitergegeben werden. 
Publication type:  DoctoralThesis 
Publishing status:  publishedVersion 
Publication date:  2022 
Keywords german:  Quantenwalks, topologische Klassifikation, diskrete Symmetrietypen 
Keywords english:  Quantum walks, topological classification, discrete symmetry types 
DDC:  530  Physik 