Note: This question is a follow up of Approximating unitary matrices.
The decompositions provided in Approximating unitary matrices are correct and worked for me without problem.
But I am now facing an issue: I want to try to optimise my circuit, and the method I want to use restricts the gate-set I can afford to use. The solution given in the previous post is no longer usable for me, because the gate-set I can use does not contain all the gates used in the solution.
My problem is the following: I need to approximate the following quantum gate $$W = \begin{pmatrix} 1&0&0&0\\ 0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0\\ 0&\frac{1}{\sqrt{2}}&\frac{-1}{\sqrt{2}}&0\\ 0&0&0&1 \\ \end{pmatrix}$$ with the gate-set $$ \left\{ H, X, R_\phi, \text{C}R_\phi, R_z, \text{C}R_z, \text{C}X, \text{CC}X \right\} $$ with $R_\phi$ defined as in Wikipedia, $R_z$ the rotation around the $Z$-axis and $\text{C}$ being a control (i.e. $\text{CC}X$ is the Toffoli gate, or double-controlled $X$ gate).
As shown by @Craig Gidney in his answer, this boils down to implement a controlled-$H$ gate with my gate-set, or equivalently a $R_y\left( \frac{\pi}{4} \right)$ gate.
I would love to have an exact decomposition, that is why I already tested:
Using the following formula $$U = e^{i\alpha} R_z(\beta) R_y(\gamma) R_z(\lambda).$$ As I only need to implement $R_y\left( \frac{\pi}{4}\right)$ I replaced $\gamma$ by $\frac{\pi}{4}$ and searched for values of $\beta$ and $\lambda$ that would result in a gate $U$ that I can implement easily with my gate-set. I did not succeed, but finding a solution to this alternative problem would solve my original problem as the equation above is equivalent to: $$R_y(\gamma) = e^{-i\alpha} R_z(-\beta) U R_z(-\lambda)$$ and $R_z$ is in my gate-set.
Trying to find a decomposition by playing with matrices, this approach did not work either, mostly because it was more random tries.
I no exact decomposition exist, I will end up using an algorithm like Solovay-Kitaev or Group Leader Optimisation, but introducing errors in this gate is a real issue as it is used extensively in all the quantum circuit.
