I have a state $$ |\tilde{\Phi_2}\rangle =\dfrac{1}{2^{3l/2}}\sum_{x=0}^{2^l-1}\sum_{y=0}^{2^l-1}\sum_{z=0}^{2^l-1}|0\rangle^{\otimes q}\otimes |x\rangle^{\otimes l}\otimes |y\rangle^{\otimes l}\otimes |z\rangle^{\otimes l} $$ Now I want to change the state $|z\rangle$ using the states $|x\rangle$ and $|y\rangle$. The controls originally $|x\rangle $ and $|y\rangle$ are originally in states $|x_0x_1...x_{l-1}\rangle$ and $|y_0x_1...y_{l-1}\rangle$, then if $$ |x_0\rangle=|y_0\rangle=|1\rangle$$ then the state $|z\rangle$ has to be changed. And the state $|z\rangle$ is changed to $|00....z_{l-3}\rangle$.
Since the $|0\rangle^{\otimes q}$ is unaltered we just operate an identity operator $I^{\otimes q}$.
The next thing I understand is that we have to break the summation into two parts where in the first part the summation indexes of $i,j$ go from $0$ to $\lfloor \dfrac{2^l-1}{2}\rfloor$.
But after that I am unable to write deduce further. Can somebody help me in writing the Dirac notation for this operator?
Edit: The transformation on the $z$ register is the right shift by $2$ operator $|z,0\rangle \to |z, \frac{z}{4}\rangle$