Hadamard (H) creates equal superposition from basis states and is fundamental to quantum algorithms.

Matrix:

$$H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$$

Action:

$$H|0\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}$$

$$H|1\rangle = \frac{|0\rangle - |1\rangle}{\sqrt{2}}$$

• Self-inverse: $H^2 = I$
• Hermitian: $H = H^\dagger$
• Eigenvalues: $+1, -1$ with eigenvectors $(|0\rangle + |1\rangle)/\sqrt{2}$ and $(|0\rangle - |1\rangle)/\sqrt{2}$
• Basis change: transforms between computational ($Z$) and Hadamard ($X$) bases

• Superposition: essential first step in quantum algorithms (Deutsch, Grover, variational)
• Basis rotation: apply before measuring in X basis to measure eigenvalue of X
• Entanglement: H followed by CNOT creates Bell states
• Bloch sphere: rotation by $\pi$ around $(x+z)/\sqrt{2}$ axis

• Matrix form: $H = \frac{1}{\sqrt{2}}(X + Z)$
• Decomposition: $H = R_X(\pi/2) R_Z(\pi)$ or alternative combinations

• Superconducting qubits: RX($\pi/2$) followed by RZ($\pi$); gate time ~40–60 ns
• Trapped ions: laser pulse combining x and z rotations
• Photonic: beam splitter (symmetric 50/50 coupler)
• Fidelity: typically 99.5–99.9%