Qubits and the Bloch Sphere

Beginner Quantum States

Created by Pavel · 11.03.2026 at 14:32 UTC

A qubit state is a normalised complex pair; global phase is invisible, relative phase steers interference. The Bloch sphere is a picture of pure single-qubit states—latitude mixes populations, longitude sets phase.

Do not confuse superposition with a classical mixture: same Z probabilities can still diverge later if phases differ. [1], [2].

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Tasks

Question 1

For $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$, what condition must hold?

Hint

Normalization uses squared magnitudes.

$|\alpha| + |\beta| = 1$

$|\alpha|^2 + |\beta|^2 = 1$

$\alpha + \beta = 1$

$\alpha^2 + \beta^2 = 0$

Question 2

If $\alpha=1/\sqrt{2}$ and $\beta=1/\sqrt{2}$, what is $P(\text{measure }1)$?

Hint

Square the amplitude magnitude of $\beta$.

0.5

0.6

0.4

Question 3

Which statement is correct about superposition?

Hint

Phase is key.

It is identical to a classical random bit.

It includes phase structure that can affect interference.

It forbids measurement outcomes 0 or 1.

It means both outcomes are observed simultaneously in one shot.

Question 4

Name the Bloch-sphere pole corresponding to $|1\rangle$.

Hint

Opposite of $|0\rangle$.

south pole

north pole

equator

it has no position

Question 5

A relative phase between amplitudes is:

Hint

Think about evolution under additional gates.

Always unobservable and never relevant

Relevant for future gate evolution/interference

Equivalent to changing the measurement device only

Forbidden for single qubits

Question 6

If $\alpha = 1/2$ and $\beta = i\sqrt{3}/2$, what is $P(\text{measure }1)$?

Hint

Use $|?eta|^2$.

0.75

0.85

0.65

1.5

Question 7

Two states differ only by a global phase factor $e^{i\phi}$. Operationally, they are:

Hint

Global phase cannot be observed directly.

Physically distinct and always measurable

Equivalent for all measurement statistics

Equivalent only in the Z basis

Not valid quantum states

Question 8

Why can two states with the same Z-basis probabilities still behave differently later?

Hint

Population equality does not imply phase equality.

Because gates ignore amplitudes

Because relative phase can alter interference under subsequent gates

Because Bloch-sphere coordinates are optional

Because normalization can be violated

Question 9

Hint

abs(beta)**2

def measure_one_probability(beta: complex) -> float:
    # TODO: |beta|^2
    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 States
Difficulty: Beginner
Completed: 0 users

Creator

Pavel