Hadamard Gate

The Hadamard gate (H) is one of the most fundamental quantum gates. It maps |0⟩ → (|0⟩ + |1⟩)/√2 (the |+⟩ state) and |1⟩ → (|0⟩ − |1⟩)/√2 (the |−⟩ state). Applied twice, it returns to the original state (H² = I). The Hadamard gate is self-inverse and unitary. In matrix form: H = (1/√2) [[1,1],[1,−1]]. It is used at the start of most quantum algorithms to create superpositions. Applying H to all n qubits in the |0...0⟩ state creates a uniform superposition over all 2ⁿ basis states simultaneously — the starting point for algorithms like Grover's search. In Bloch sphere terms, H rotates the Bloch vector by 180° about the axis halfway between X and Z.