Interference is the feature: why amplitudes (not probabilities) matter

If you try to understand quantum algorithms as “probabilities evolving,” you’ll miss the point.

Quantum circuits evolve amplitudes, which are complex numbers. Probabilities only appear at the very end when you measure.

Two paths, one outcome

Imagine two “computational paths” that both lead to the same basis state. If their amplitudes are:

• Path A contributes (+0.5)
• Path B contributes (+0.5)

Then the total amplitude is (1.0) and the probability is (|1.0|^2 = 1).

Now flip the phase of one path:

• Path A contributes (+0.5)
• Path B contributes (-0.5)

Total amplitude is (0), so the probability becomes (|0|^2 = 0). The outcome is eliminated.

That’s interference.

Why phase matters (even though you never observe it directly)

In a quantum program you often do this loop:

1. Spread amplitude over many possibilities (superposition)
2. Apply an oracle-like operation that flips phases on “good” states
3. Apply a mixing transform that converts phase differences into amplitude differences
4. Measure

That’s the shape of Grover’s algorithm and appears in many other places (phase estimation, amplitude amplification variants, etc.).

The big intuition

• Classical randomness can only reweight probabilities.
• Quantum dynamics can reweight probabilities by manipulating phases first.

If you want a “one-liner” mental model for quantum speedups:

Quantum algorithms work when you can compute something in phase and then turn phase into amplitude.

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