  Holographic complexity proposals have sparked interest in quantifying the
cost of state preparation in quantum field theories and its possible dual
gravitational manifestations. The most basic ingredient in defining complexity
is the notion of a class of circuits that, when acting on a given reference
state, all produce a desired target state. In the present work we build on
studies of circuits performing local conformal transformations in general
two-dimensional conformal field theories and construct the exact gravity dual
to such circuits. In our approach to holographic complexity, the gravity dual
to the optimal circuit is the one that minimizes an externally chosen cost
assigned to each circuit. Our results provide a basis for studying exact
gravity duals to circuit costs from first principles.
