Probability Amplitudes Without Scary Math
An amplitude is a signed or complex number attached to an outcome; the outcome's probability is the amplitude's squared magnitude, and a valid state's probabilities sum to exactly 1. Squaring erases sign information — phase — which later gates can still exploit.
What is an amplitude, if not a probability?
A probability is a real number between 0 and 1. An amplitude is the number a quantum state attaches to an outcome, and it may be negative or complex. The bridge between them is one operation: probability = squared magnitude of the amplitude.
Worked example. The amplitude −1/√2 ≈ −0.7071 gives probability (−0.7071) × (−0.7071) = 0.5. The minus sign vanishes from the probability — but it is not meaningless. It records phase: the sign relationship between this amplitude and the others, which later gates can exploit.
How do you square a complex amplitude?
A complex number has the form a + bi, where i is the imaginary unit (i × i = −1). Its squared magnitude needs no complex analysis: square each part and add them, giving a² + b².
Worked example. For the amplitude 0.3 + 0.4i: 0.3 × 0.3 = 0.09 and 0.4 × 0.4 = 0.16, so the probability is 0.09 + 0.16 = 0.25 — a 25% chance.
Validity check. Pair that amplitude with a second amplitude 0.8660 (that is √0.75). The totals are 0.25 + 0.75 = 1.00 — a valid state.
How do you check and repair normalisation?
Rule: sum the squared magnitudes of all amplitudes. A valid state sums to exactly 1 — that property is called normalisation.
- (0.6, 0.8): 0.36 + 0.64 = 1.00 — valid.
- (0.7071, −0.7071): 0.5 + 0.5 = 1.00 — valid; signs do not affect the check.
- (1, 1): 1 + 1 = 2 — invalid as written.
To repair a non-zero invalid vector, divide every amplitude by the square root of the sum. For (1, 1): √2 ≈ 1.4142, giving (0.7071, 0.7071), which now sums to 1.
In your own tooling, surface normalisation failures as errors rather than silently clamping or renormalising — silent repair hides bugs upstream.
Can you choose any probability you like?
Yes — with a rotation gate. The name comes from a geometric picture of the qubit that lesson 5 introduces; for now only the formula matters. RY(θ) takes |0⟩ to the state cos(θ/2)|0⟩ + sin(θ/2)|1⟩, where the parameter θ is an angle in radians.
Worked example. Set θ = π/3 ≈ 1.0472. Then θ/2 = π/6, so cos(π/6) = 0.8660 and sin(π/6) = 0.5. Squaring: 0.8660 × 0.8660 = 0.75 and 0.5 × 0.5 = 0.25. Ideal outcome: 75% zeros, 25% ones.
Run it: dial in a 75/25 split
Where does the phase information go?
Compare |+⟩, amplitudes (0.7071, 0.7071), with |−⟩, amplitudes (0.7071, −0.7071). Both give 50/50 in this measurement, because squaring erases the sign.
A sign or complex-factor difference between amplitudes is relative phase. It is real information: later gates can turn it into different probabilities — lesson 7 does exactly this.
Multiplying every amplitude by the same unit-size factor — for example turning (0.7071, −0.7071) into (−0.7071, 0.7071) by multiplying the whole state by −1 — is global phase. No measurement, now or later, can detect global phase on an isolated state.
Does amplitude 0.5 mean a 50% chance?
No. A real amplitude of 0.5 gives probability 0.5 × 0.5 = 0.25 — a 25% chance.
The amplitude that produces a 50% chance is 1/√2 ≈ 0.7071, because 0.7071 × 0.7071 = 0.5. Keep the two columns separate: amplitudes are internal, signed or complex; probabilities are their squared magnitudes.