# Hadamard Gate

**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: 

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

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

## Properties

- **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

## Uses

- **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

## Composition

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

## Implementation

- **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%