The Hadamard gate, visually

Drag the input arrow. Trace each amplitude through the H gate to see how it splits and combines at the outputs.

input
|0> |1>
amplitudes
probabilities
P(|0>)1.00
P(|1>)0.00
α β H ×1/√2 ×1/√2 ×1/√2 ×−1/√2 +0.71 +0.00 +0.71 −0.00 α' β'
after H
|0> |1>
amplitudes
probabilities
P(|0>)0.50
P(|1>)0.50

Each input amplitude splits through H into both outputs. The solid lines show the "straight" paths (α to α', β to β'); the dashed lines show the crossing paths (α to β', β to α'). The coefficients are fixed: ×1/√2 everywhere except β to β', which gets ×−1/√2 (the minus sign). At each output, two contributions arrive and add. The values near the output dots show the pre-interference contributions; the output amplitude bars show their sum.