# Can W gate be written only using H,T?

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)?

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 $$Y\mapsto -Y$$.