Amplitudes and probability: squaring

The square of an amplitude is a probability. Amplitudes can be negative and can cancel when they add up; this is something probabilities cannot do, and it is the secret of quantum interference.

Probability = amplitude squared

If a result has amplitude a, the probability of measuring it is a². A result with amplitude 1 is certain (1² = 1); one with amplitude 0 never appears. In an equal superposition the amplitudes are 1/√2 ≈ 0.707, and 0.707² ≈ 0.5, that is 50%.

Why bother with amplitudes instead of writing probabilities directly? Because amplitudes have a property probabilities lack: they can be negative.

A negative amplitude

Consider the state (0.707, −0.707). The squares are still 0.5 and 0.5: measured, it behaves exactly like a coin toss, 50–50. A measurement cannot see the minus sign.

But the two states are different. In the playground you get this state by applying X and then H; applying only H gives (0.707, 0.707). The difference shows up at the next gate.

Amplitudes add up and can cancel

In a quantum circuit a result can be reached along several paths, and the amplitudes of those paths add up. If two paths carry +0.5 and −0.5, the total is 0: that result never appears. This is called destructive interference. Probabilities cannot do this, because they are never negative and only grow when added.

This is exactly what happens with two H gates in a row. The two paths to 1 carry +½ and −½ and cancel; the paths to 0 reinforce to ½ + ½ = 1. Quantum algorithms use this to switch off wrong answers and amplify the right one.

Try it yourself

In the negative amplitude example open the state vector: the amplitude of |1⟩ is −0.707, yet the probabilities are 50–50. Then add one more H at the end: X, H, H always gives 1, because applying H twice undoes it and only the X remains.

Open the circuit in your browser

The circuit playground is in Turkish. Button guide: “1000 kez ölç” = measure 1000 times, “Durum vektörünü göster” = show the state vector, “Bu devrenin Qiskit kodu” = this circuit’s Qiskit code, “Geri al” = undo, “Temizle” = clear. Click a gate in the circuit to remove it.

Key points

The probability of a result is the square of its amplitude.
Amplitudes can be negative; a measurement alone cannot see the sign.
Amplitudes of paths to the same result add up and can cancel (destructive interference).
Quantum algorithms switch off wrong answers with interference and amplify the right one.

Test yourself

If a result has amplitude −0.6, what is the probability of measuring it?

−0.36
0.36
0.6

Show the answer

0.36. The probability is the amplitude squared: (−0.6)² = 0.36. A square is never negative, so a probability never is either.

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