  We investigate the first law of complexity proposed in arXiv:1903.04511,
i.e., the variation of complexity when the target state is perturbed, in more
detail. Based on Nielsen's geometric approach to quantum circuit complexity, we
find the variation only depends on the end of the optimal circuit. We apply the
first law to gain new insights into the quantum circuits and complexity models
underlying holographic complexity. In particular, we examine the variation of
the holographic complexity for both the complexity=action and complexity=volume
conjectures in perturbing the AdS vacuum with coherent state excitations of a
free scalar field. We also examine the variations of circuit complexity
produced by the same excitations for the free scalar field theory in a fixed
AdS background. In this case, our work extends the existing treatment of
Gaussian coherent states to properly include the time dependence of the
complexity variation. We comment on the similarities and differences of the
holographic and QFT results.
