  There are two elementary superconducting qubit types that derive directly
from the quantum harmonic oscillator. In one the inductor is replaced by a
nonlinear Josephson junction to realize the widely used charge qubits with a
compact phase variable and a discrete charge wavefunction. In the other the
junction is added in parallel, which gives rise to an extended phase variable,
continuous wavefunctions and a rich energy level structure due to the loop
topology. While the corresponding rf-SQUID Hamiltonian was introduced as a
quadratic, quasi-1D potential approximation to describe the fluxonium qubit
implemented with long Josephson junction arrays, in this work we implement it
directly using a linear superinductor formed by a single uninterrupted aluminum
wire. We present a large variety of qubits all stemming from the same circuit
but with drastically different characteristic energy scales. This includes flux
and fluxonium qubits but also the recently introduced quasi-charge qubit with
strongly enhanced zero point phase fluctuations and a heavily suppressed flux
dispersion. The use of a geometric inductor results in high precision of the
inductive and capacitive energy as guaranteed by top-down lithography - a key
ingredient for intrinsically protected superconducting qubits. The geometric
fluxonium also exhibits a large magnetic dipole, which renders it an
interesting new candidate for quantum sensing applications.
