It is known that the complexity class P is equivalent to the class of problems decided by polynomial-time uniform familiy of circuits. When stating the complexity of algorithms as this family of circuits, this complexity is usually stated in terms of the depth and size of the circuit. What I have not understood well is why the authors don't include the complexity of the Turing machine constructing the circuits as well. This seems an important part of the overall algorithm. Could somebody clear this up for me?
In particular im interested in this question in the context of quantum circuits. When implementing a quantum algorithm usually this is expressed as a quantum circuit, and quantum algorithms papers report the complexity in terms of the depth. Why not consider the complexity of the classical computer that has to construct the circuit in the first place? Can the complexity of contructing the circuit be bigger than that of running the circuit itself?