# Is there any shortcut to create a controlled gate or any technique?

I was solving Controlled H gate from qiskit textbook -> Basic Circuit Identities

There I tried to find the unitary matrix for CH gate but it get's really complicated when I use tensor product or other stuff. Is there any shortcut to prove that the given circuit is the equivalent to CH gate --- Circuit is --> The shortcut is to think about what happens to $$q_1$$ in the two separate cases of $$q_0$$ being either $$|0\rangle$$ or $$|1\rangle$$. If it works for both of those, then by linearity it works on all inputs.
So, if $$q_0$$ is in $$|0\rangle$$, the controlled-not does nothing. Effectively,the only gates that act on $$q_1$$ are two single-qubit gates which are the inverse of each other. Hence, the net effect is the identity.
If $$q_0$$ is in $$|1\rangle$$, the controlled-not applies $$X$$ on $$q_1$$. Hence, the net sequence is $$R_Y(-\pi/4)XR_Y(\pi/4).$$ You could just multiply this out. Alternatively, use commutation relations: $$R_Y(\theta)X=XR_Y(-\theta).$$ Thus, the net gate is $$XR_Y(\pi/2)=X(I-iY)/\sqrt{2}=(X+Z)/\sqrt{2}.$$
Your overall description is "if $$q_0$$ is in 0, do nothing, otherwise apply Hadamard". In other words, controlled-Hadamard.