Suppose I am working with a class of states which consist of a superposition of $O(\text{poly}(N))$ computational basis states on $N$ qubits. Examples of this would be the subspace of states of fixed Hamming weight $k < N$, or alternatively CISD states in quantum chemistry. If my goal is to implement arbitrary polynomial-sized states with this restriction, can I assume that there is a procedure to implement such states with a quantum circuit that also scales polynomially in $N$?
A sort-of brute force idea I had for how this could be done is to build each term in the superposition one by one. If each term can be added one-by-one with a polynomial circuit, then implementing the total state can also be done with a polynomial circuit. But I am not sure if this can be done in practice.