Creates equal superposition from a basis state.
The Hadamard gate is often the first gate you encounter in quantum computing, and for good reason. It takes a qubit that is definitely in |0⟩ or |1⟩ and puts it into an equal superposition of both possibilities. This is the fundamental operation that separates quantum computing from classical computing — it creates the parallelism that quantum algorithms exploit. Think of it as opening a door: before Hadamard, your qubit is locked into one value. After Hadamard, both values exist simultaneously as amplitudes in the quantum state, ready for interference and entanglement.
Hadamard is a unitary, Hermitian, and involutory operator (H² = I). It maps the computational basis to the diagonal basis: |0⟩ → |+⟩ = (|0⟩ + |1⟩)/√2 and |1⟩ → |−⟩ = (|0⟩ − |1⟩)/√2. Geometrically, it is a 180° rotation around the axis halfway between X and Z on the Bloch sphere. The key insight is that H creates states with definite relative phase (±0), and that phase carries information even though the measurement probabilities look identical for |+⟩ and |−⟩ in the computational basis.
Measurement odds spread to roughly 50/50.
The relative phase (± sign) carries information that a later Hadamard or interference step can reveal.
Rotates 180° around the axis halfway between X and Z. Maps the north pole (|0⟩) to the equator (|+⟩) and the south pole (|1⟩) to the opposite equatorial point (|−⟩).
Hadamard does not make a qubit “both 0 and 1 at once” in any classical sense. It creates a superposition where both outcomes have well-defined amplitudes, but each measurement still produces a single definite result.
Without Hadamard, quantum circuits would just be classical permutations. It is the gateway to superposition, and superposition is what makes interference and quantum speedups possible.
You apply H to |0⟩ and then apply H again. What state do you get?