Hadamard Gate & Superposition

Hadamard Gate & Superposition

The Hadamard gate, written H, is the gate that makes quantum computing look quantum. It turns definite classical states into equal superpositions, it turns superpositions back into definite states, and it is the workhorse behind Deutsch-Jozsa, Grover, quantum phase estimation, and BB84. If you understand H, you understand superposition.

The Matrix and What It Does

H = (1/√2) · [[1,  1],
              [1, −1]]

Applied to the computational basis states:

Measured in the Z basis, both |+⟩ and |−⟩ yield 0 and 1 with probability 1/2 each. What distinguishes them is phase, and that phase becomes visible only when you later apply another gate (like another H).

Geometrically, H is a 180° rotation of the Bloch sphere about the diagonal axis (X+Z)/√2. That rotation swaps the north pole with the +X point, which is exactly "turning |0⟩ into |+⟩".

Interference: Why Two H's Is Nothing

Because H is self-inverse, H·H = I. If you apply H twice to |0⟩, you get back exactly |0⟩:

H|0⟩ = (|0⟩ + |1⟩)/√2
H·H|0⟩ = H·(|0⟩ + |1⟩)/√2
       = (H|0⟩ + H|1⟩)/√2
       = ((|0⟩ + |1⟩)/√2 + (|0⟩ − |1⟩)/√2)/√2
       = |0⟩

The |1⟩ contributions destructively interfere and cancel, while the |0⟩ contributions constructively interfere and return with amplitude 1.

[animation: interference] This is the whole game: quantum algorithms arrange amplitudes to cancel on wrong answers and reinforce on right ones. H is the simplest place to see it happen.