  This thesis is set in the framework of superconducting transmon-type qubit
architectures with special focus on two important types of coupling between
qubits and harmonic resonators: transverse and longitudinal coupling. We will
see that longitudinal coupling offers some remarkable advantages with respect
to scalability and readout. This thesis will focus on a design, which combines
both these coupling types in a single circuit and provides the possibility to
choose between pure transverse and pure longitudinal or have both at the same
time. We will start with an introduction to circuit quantization, where we will
explain how to describe and analyze superconducting electrical circuits in a
systematic way and discuss which characteristic circuit elements make up qubits
and resonators. We will then introduce the two types of coupling between qubit
and resonator which are provided in our design. Translating this discussion
from the Hamiltonian level to the language of circuit quantization, we will
show how to design circuits with specifically tailored couplings. We will focus
on our circuit design that consists of an inductively shunted transmon qubit
with tunable coupling to an embedded harmonic mode. The distinctive feature of
the tunable design is that the transverse coupling disappears when the
longitudinal is maximal and vice versa. Subsequently, we will turn to the
implementation of our circuit design, discuss how to choose the parameters, and
present an adapted alternative circuit, where coupling strength and
anharmonicity scale better than in the original circuit. In addition, we
present a proposal for an experimental device that will serve as a prototype
for a first experiment. We will conclude the thesis discussing different
possibilities to do readout with our circuit design, including a short
discussion of the influence of dissipation.
