  Using the volume proposal, we compute the change of complexity of holographic
states caused by a small conformal transformation in AdS$_{3}$/CFT$_{2}$. This
computation is done perturbatively to second order. We give a general result
and discuss some of its properties. As operators generating such conformal
transformations can be explicitly constructed in CFT terms, these results allow
for a comparison between holographic methods of defining and computing
computational complexity and purely field-theoretic proposals. A comparison of
our results to one such proposal is given.
