Hadamard gate

Hadamard is the single-qubit unitary $H = \frac{1}{\sqrt{2}}\begin{pmatrix}1 & 1\\ 1 & -1\end{pmatrix}$. It sends $|0\rangle \mapsto \frac{1}{\sqrt2}(|0\rangle+|1\rangle)$ and $|1\rangle \mapsto \frac{1}{\sqrt2}(|0\rangle-|1\rangle)$, so a computational-basis measurement always returns $0$ or $1$ with probability $1/2$ regardless of which state it acted on — the difference is the relative sign in front of $|1\rangle$, a phase that only becomes visible once another gate lets the two branches interfere. $H$ is both Hermitian and unitary, so $H^\dagger = H = H^{-1}$, and direct matrix multiplication gives $H^2 = I$, exactly the identity this record's matrix check confirms. Geometrically $H$ is a $\pi$ rotation of the Bloch sphere about the axis $(\hat x + \hat z)/\sqrt2$, exchanging the $Z$ and $X$ eigenbases. It is the standard way to open superposition-based circuits such as Bell states, GHZ states, Grover diffusion, and the QFT.