Can W gate be written only using H,T?

Question

I want to write the decompose the gate $W=(X+Y)/\sqrt{2}$ using only $H$ and $T$ (and all the derived Clifford gates basically). I know $H=(X+Z)/\sqrt{2}$ is it possible to obtain exactly $W$ from this set (no approximation)?

Answer 1

Try the sequence: $$ HT^6HT^2H. $$ What was my thinking? I'm used to doing a transformation that looks something like $$ S^\dagger H S. $$ The action is the $S$ is to preserve the $Z$ term inside $H$, but transforms the $X$ term into a $Y$. So, we take the $S=T^2$ and $S^\dagger=T^6$, and this gives $$ T^6HT^2=(Z-Y)/\sqrt{2}. $$ Now, conjugate with Hadamards to change $Z\mapsto X$ and $Y\mapsto -Y$.