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According to the pioneering work of Nielsen and collaborators, the length of
the minimal geodesic in a geometric realization of a suitable operator space
provides a measure of the quantum complexity of an operation. Compared with the
original concept of complexity based on the minimal number of gates required to
construct the desired operation as a product, this geometrical approach amounts
to a more concrete and computable definition, but its evaluation is nontrivial
in systems with a high-dimensional Hilbert space. The geometrical formulation
can more easily be evaluated by considering the geometry associated with a
suitable finite-dimensional group generated by a small number of relevant
operators of the system. In this way, the method has been applied in particular
to the harmonic oscillator, which is also of interest in the present paper.
However, subtle and previously unrecognized issues of group theory can lead to
unforeseen complications, motivating a new formulation that remains on the
level of the underlying Lie algebras for most of the required steps. Novel
insights about complexity can thereby be found in a low-dimensional setting,
with the potential of systematic extensions to higher dimensions as well as
interactions. Specific examples include the quantum complexity of various
target unitary operators associated with a harmonic oscillator, inverted
harmonic oscillator, and coupled harmonic oscillators. The generality of this
approach is demonstrated by an application to an anharmonic oscillator with a
cubic term.