Bachelor and diploma theses

The tensor-product states (TPS) are frequently used in modern physics in order to obtain energies and eigenstates of quantum models. The typical systems to be studied are spin models whose Hamiltonians describe magnetism at zero temperature. By means of the tensor-product states, a quantum wave function (eigenstate) is approximated. The wave function contains both the physical degrees of the freedom, describing the Hilbert space, and the auxiliary (non-physical) degrees of the freedom, whose size controls range of the correlations, i.e., the accuracy. Many-body interacting spin systems are to be characterized by the TPS at the phase transition be means of the quantum entanglement. Since it is a non-trivial task, being non-analytically solvable, numerical approaches can be used, in particular, an arbitrary programming language can be freely applied, e.g., Python, MatLab, C++, etc.

The tensor networks describes a specific mathematical approach in quantum mechanics, which enables to calculate (approximate) quantum ground states accurately. First, a matrix-product state method is explained, followed by its application to such quantum spin systems, whose Hamiltonians can describe magnetic properties of materials, magnetic spin orderings, including quantum phase transitions. An arbitrary programming language can be used for the study, e.g., Python, MatLab, C++. etc.

Non-Euclidean geometry exhibits a non-zero Gaussian curvature. An iterative numerical algorithm is to be proposed by renormalization-group methods for curved networks with negative curvatures. These networks mimic interaction structure of spin models, which describe physical systems in terms of Hamiltonians. Since such systems are not exactly solvable, numerical techniques need to be employed, so that an arbitrary programming language can be used, e.g., Python, MatLab, C++. etc. As the outcome, the specification of various universality classes of the spin models is to be given, including the elucidation of the mechanism, which breaks long-order correlations at phase transition on the negatively curved geometries. Single- and two-particle correlations, including entanglement entropy are to be calculated.

The adiabatic theorem tells us that evolving an eigenstate with a weakly time-dependent Hamiltonian results in a state near an instantaneous eigenstate, if some conditions are met. In this diploma thesis, we will first analyze these conditions in some depth. Second, we will look for computational applications of such processes, and their generalizations (quantum annealing). We will focus on stoquastic Hamiltonians, in particular those related to continuous-time quantum walks.