  We obtain the complexity geometry associated with the Hamiltonian of a
quantum mechanical system, specifically in cases where the Hamiltonian is
explicitly time-dependent. Using Nielsen's geometric formulation of circuit
complexity, we calculate the bi-invariant cost associated with these
time-dependent Hamiltonians by suitably regularising their norms and obtain
analytical expressions of the costs for several well-known time-dependent
quantum mechanical systems. Specifically, we show that an equivalence exists
between the total costs of obtaining an operator through time evolution
generated by a unit mass harmonic oscillator whose frequency depends on time,
and a harmonic oscillator whose both mass and frequency are functions of time.
These results are illustrated with several examples, including a specific
smooth quench protocol where the comparison of time variation of the cost with
other information theoretic quantities, such as the Shannon entropy, is
discussed.
