Hadamard Gates
The Hadamard gate is a fundamental single-qubit gate in quantum computing. It transforms basis states into superpositions, enabling interference effects that are central to quantum algorithms.
Action on States:
- |0⟩ → (|0⟩ + |1⟩)/√2 = |+⟩
- |1⟩ → (|0⟩ - |1⟩)/√2 = |−⟩
Circuit: q0 → H
Figure A: Hadamard gate creates superposition - measurement gives random 0 or 1 with 50% probability each. This randomness is fundamental to quantum mechanics, not just hidden information. The qubit genuinely exists in both states simultaneously until measured.
Ready to run - Hadamard creates superposition, measurement gives random 0 or 1
Circuit: |0⟩ → H → H
Figure B: Two Hadamard gates in sequence cancel each other out - the output always equals the input. H(H(|ψ⟩)) = |ψ⟩
Ready to run - Two Hadamard gates in sequence return the qubit to its original state
Bloch Sphere Representation
The Hadamard gate rotates |0⟩ from the north pole to the equator at |+⟩ = (|0⟩ ± |1⟩)/√2