Interference happens when amplitudes combine, turning a hidden phase difference into a visible change in measurement probabilities.
Without interference, superposition would only look like randomness. Interference is what converts superposition into a computational resource. Every quantum speedup relies on making useful outcomes interfere constructively and useless ones interfere destructively.
Two possibilities can reinforce each other (constructive interference) or cancel each other (destructive interference). This is what you saw in the double-slit experiment, and the same thing happens in circuits. A Z gate changes the phase of a qubit without changing its measurement probabilities. But if you then apply a Hadamard gate, the amplitudes combine and the phase difference becomes a measurable change. This is the core mechanism that makes quantum algorithms work.
Water waves can meet crest-to-crest (they add up) or crest-to-trough (they cancel). Quantum amplitudes work similarly -- they add or cancel depending on their phases. The analogy is useful for the add-or-cancel idea, but remember: quantum amplitudes are complex numbers, not physical waves.
The Z gate flips the sign of the amplitude. In the standard measurement basis, this does not change the probabilities. But a subsequent Hadamard mixes the and amplitudes, and the sign difference creates constructive interference on one outcome and destructive interference on the other. The identity shows that a hidden phase change, sandwiched between two basis changes, produces a visible bit flip.
Open the simulator and see this concept in action. Watch how the state changes and compare it to what you just learned.
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