HZH = X and Friends

The Hadamard gate exchanges the computational ($Z$) basis and the $X$ basis. A useful consequence is that conjugating $Z$ by $H$ turns it into $X$:

$H Z H = X.$

You can check it by multiplying the matrices, but the idea is cleaner: $Z$ flips the sign of $|1\rangle$ (a reflection about the $z$-axis), while $H$ swaps the $x$- and $z$-axes. Doing "swap axes → reflect about $z$ → swap back" is the same as reflecting about $x$, which is exactly what $X$ does.

This is your first taste of building one operation out of others — the heart of compiling quantum programs onto a fixed gate set.

Try it

Implement $X$ using only $H$ and $Z$. This exercise grades the whole operation (its unitary matrix), so doing nothing won't pass — you must actually realize $X$.

Sign in on the full site to ask questions and join the discussion.