Single-Qubit Gates

H Then H Interference

recombining amplitudes

Apply H to the equal superposition and watch the two branches recombine instead of treating them as hidden coin flips. In this pinned toy ledger the zero amplitude builds as 1/2 plus 1/2 while the one amplitude cancels as 1/2 minus 1/2; real gate checks are limited by phase control, noise, decoherence, and readout statistics.

Applying H again shows constructive and destructive interference in a finite qubit state. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

A second H recombines amplitudes

The checked H output from the equal superposition is the zero basis state.

H∣+⟩=∣0⟩H\lvert + \rangle = \lvert 0\rangle
H then HThe second H restores the zero basis state.state1/sqrt(2) zero1/sqrt(2) onestate1 zero0 oneHgate

The zero outcome is constructive

For zero, the two path contributions are 1/2 plus 1/2, giving amplitude 1.

12+12=1\frac{1}{2} + \frac{1}{2} = 1
Constructive amplitudeExact path amplitudes add to one.1/2 + 1/2= 1P = 1amplitude sum

The one outcome cancels

For one, the two path contributions are 1/2 plus negative 1/2, giving amplitude 0.

12−12=0\frac{1}{2} - \frac{1}{2} = 0
Destructive amplitudeExact path amplitudes cancel.1/2 + -1/2= 0P = 0amplitude sum