  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.
