From a 9×9 Hamiltonian lying 9D space, I choose a certain subspace of 4D for designing a two qubit gate. Now the original unitary time evolution operator also lies in 9D space and it's a 9×9 size matrix. For action of unitary time evolution operator on the two qubit gate made out of 4D subspace it is required to project the unitary time evolution operator in the 4D subspace. After reviewing literature, I came across an article doing same thing with the use of projection operator.
My question- How to find the projection operator on the subspace?
Also I guess projection operator will be 4×4 matrix, so how will it act on the unitary time evolution operator which is a 9×9 matrix.
P.S.- I took the definition of projection operator from "Quantum Computation and Quantum Information, Isaac Chuang and Michael Nielsen".