  In Nielsen's geometric approach to quantum complexity, the introduction of a
suitable geometrical space, based on the Lie group formed by fundamental
operators, facilitates the identification of complexity through geodesic
distance in the group manifold. Earlier work had shown that the computation of
geodesic distance can be challenging for Lie groups relevant to harmonic
oscillators. Here, this problem is approached by working to leading order in an
expansion by the structure constants of the Lie group. An explicit formula for
an upper bound on the quantum complexity of a harmonic oscillator Hamiltonian
with time-dependent frequency is derived. Applied to a massless test scalar
field on a cosmological de Sitter background, the upper bound on complexity as
a function of the scale factor exhibits a logarithmic increase on super-Hubble
scales. This result aligns with the gate complexity and earlier studies of de
Sitter complexity. It provides a proof of concept for the application of
Nielsen complexity in cosmology, together with a systematic setting in which
higher-order terms can be included.
