Real-time simulations of transmon systems with time-dependent Hamiltonian models

In this thesis we study aspects of Hamiltonian models which can affect the time evolution of transmon systems. We model the time evolution of various systems as a unitary real-time process by numerically solving the time-dependent Schr\"odinger equation (TDSE). We denote the corresponding computer models as non-ideal gate-based quantum computer (NIGQC) models since transmons are usually used as transmon qubits in superconducting prototype gate-based quantum computers (PGQCs).We first review the ideal gate-based quantum computer (IGQC) model and provide a distinction between the IGQC, PGQCs and the NIGQC models we consider in this thesis. Then, we derive the circuit Hamiltonians which generate the dynamics of fixed-frequency and flux-tunable transmons. Furthermore, we also provide clear and concise derivations of effective Hamiltonians for both types of transmons. We use the circuit and effective Hamiltonians we derived to define two many-particle Hamiltonians, namely a circuit and an associated effective Hamiltonian. The interactions between the different subsystems are modelled as dipole-dipole interactions. Next, we develop two product-formula algorithms which solve the TDSE for the Hamiltonians we defined. Afterwards, we use these algorithms to investigate how various frequently applied assumptions affect the time evolution of transmon systems modelled with the many-particle effective Hamiltonian when a control pulse is applied. Here we also compare the time evolutions generated by the effective and circuit Hamiltonian. We find that the assumptions we investigate can substantially affect the time evolution of the probability amplitudes we model. Next, we investigate how susceptible gate-error quantifiers are to assumptions which make up the NIGQC model. We find that the assumptions we consider clearly affect gate-error quantifiers like the diamond distance and the average infidelity.