Measurement and Born Rule

Beginner Quantum Measurement

Created by Pavel · 11.03.2026 at 14:32 UTC

You prepared a superposition and measured once: the histogram you see is not the wavefunction, it is a sample from the distribution the Born rule defines. Squared magnitudes turn amplitudes into probabilities, so one shot can land anywhere the distribution allows, and only many shots reveal the underlying weights.

The catch is reporting: a point estimate without uncertainty overstates confidence, especially when counts are small. Changing the measurement basis—often implemented as a rotation before readout—is literally changing which operator you are estimating, not just a visual tweak. Wikipedia gives a compact statement of the rule [1].

Sources

[1]https://en.wikipedia.org/wiki/Born_rule Return to text

University approvals: 0

Tasks

Question 1

Born rule probabilities are computed from:

Hint

Magnitude squared.

Raw amplitudes without magnitude

Squared magnitudes of amplitudes

Only real parts of amplitudes

Only imaginary parts of amplitudes

Question 2

Why do we run many measurement shots?

Hint

Single shot is random.

to estimate the probability distribution

none of the above

all of the above

insufficient information

Question 3

Changing basis before measurement is equivalent to:

Hint

Basis determines what property is probed.

Changing the question/observable being measured

Destroying all information

Forcing deterministic outcomes

Eliminating shot noise

Question 4

If a qubit is in $|0\rangle$, what is $P(\text{measure }1)$ in computational basis?

Hint

Orthogonal basis states.

0.25

0.5

Question 5

A rigorous measurement report should include:

Hint

Think reproducible experimental reporting.

Only the best single-shot outcome

Estimated statistics and uncertainty

No mention of shots

Only circuit drawing

Question 6

You observed 230 outcomes of '1' in 1000 shots. What is the estimated P(1)?

Hint

Count divided by total shots.

0.23

0.33

0.13

0.46

Question 7

If shot count N increases by 4x, the typical standard error of estimated probabilities scales approximately by:

Hint

Sampling error scales as $1/\sqrt{N}$.

2x larger

unchanged

about 2x smaller

about 4x smaller

Question 8

Why are pre-rotations before computational-basis readout important?

Hint

Basis choice determines the measured question.

They only improve plotting aesthetics

They map different observables into the measured basis

They remove all stochasticity from measurement

They guarantee zero hardware noise

Question 9

Implement born_probability(amplitude: complex) -> float returning the Born probability $|z|^2$ for a single amplitude $z$. Use abs and not numpy (stdlib only).

Tests run in expression mode on the default sandbox.

Hint

For complex z, |z|2 == (z.real2 + z.imag**2).

def born_probability(amplitude: complex) -> float:
    # TODO: return |amplitude|^2 as float
    pass

Starter code is prefilled; replace TODO blocks with your solution.

Runtime output (stdout/stderr)

2 test cases will be used for grading

Run checks runtime behavior only. Final correctness is evaluated when you submit.

Card Info

Topic: Quantum Measurement
Difficulty: Beginner
Completed: 0 users

Creator

Pavel