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Using the volume proposal, we compute the change of complexity of holographicstates caused by a small conformal transformation in AdS$_{3}$/CFT$_{2}$. Thiscomputation is done perturbatively to second order. We give a general resultand discuss some of its properties. As operators generating such conformaltransformations can be explicitly constructed in CFT terms, these results allowfor a comparison between holographic methods of defining and computingcomputational complexity and purely field-theoretic proposals. A comparison ofour results to one such proposal is given.

Complexity change under conformal transformations in AdS$_{3}$/CFT$_{2}$