  In this paper, we study circuit complexity in Proca theory with Nielsen's
approach and Fubini-Study (FS) metric approach. We place the fields on a
lattice to gain a regularized theory, and obtain the ground state by adopting
proper coordinates. We calculate complexities of the ground and thermofield
double (TFD) states with Nielsen's approach, complexity of the TFD state is
found to grows like a logarithmic function. We quantize the Proca fields and
give the approximate ground state and TFD state by acting unitary circuit
operators on the associated reference states. The circuit lengths are
calculated with FS metric, the minimal lengths are given according to the
associated geometric spaces. The complexity of TFD state is found to grows
linearly with time.
