Puts a qubit into an equal superposition of and .
The Hadamard gate takes a qubit that is definitely or and splits it into an equal superposition of both. A superposition means the qubit's state is a weighted combination of and at the same time, with each branch carrying a number called an amplitude. Measuring a superposition gives 0 or 1 with probabilities determined by those amplitudes. Before Hadamard, your qubit is locked into one value. After Hadamard, both values exist as amplitudes, and interference between them is what gives quantum algorithms their power.
Hadamard is unitary (it preserves total probability), Hermitian (it equals its own conjugate transpose, so it is also a valid measurement observable), and involutory (applying it twice gives the identity: H² = I). It maps the computational basis states to the diagonal basis: → and → |−⟩ = ( − . On the Bloch sphere (a 3D representation where every single-qubit state maps to a point on a unit sphere), Hadamard is a 180° rotation around the axis halfway between X and Z. The states and |−⟩ differ only in their relative phase (the sign between the and terms). That sign is invisible when you measure in the computational basis, because both states give 50/50 outcomes. But the sign carries information that a later Hadamard or interference step can extract.
Measurement outcomes spread to roughly 50/50 when starting from or .
The relative phase (± sign between the and amplitudes) encodes information that a later Hadamard or interference step can reveal.
Rotates the Bloch vector 180° around the axis halfway between X and Z. The north pole moves to the +X equator . The south pole moves to the −X equator (|−⟩).
It is tempting to think Hadamard makes a qubit "both 0 and 1 at once" in a classical sense. That sounds plausible because measurement gives either outcome. What actually happens is that the qubit enters a superposition where both outcomes have well-defined amplitudes, but each individual measurement still produces a single definite result.
Without Hadamard, quantum circuits would just shuffle definite states around, no differently from classical logic. Hadamard opens the door to superposition, and superposition enables interference, which is the mechanism behind every known quantum speedup.
You apply H to |0⟩ and then apply H again. What state do you get?