Self-Assessment Quiz: The Qubit

Twenty questions on superposition, the Bloch sphere, phase, normalization, and what a qubit actually is. Answer each before opening the key. Aim for 16+.

Question 1

A general single-qubit state is $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$. The normalization condition is:

A) $\alpha + \beta = 1$ B) $|\alpha| + |\beta| = 1$ C) $|\alpha|^2 + |\beta|^2 = 1$ D) $\alpha^2 + \beta^2 = 1$

Question 2

For $|\psi\rangle = \frac{1}{\sqrt{2}}|0\rangle - \frac{1}{\sqrt{2}}|1\rangle$, the probability of measuring $0$ in the computational basis is:

A) $\tfrac{1}{\sqrt 2}$ B) $\tfrac12$ C) $-\tfrac12$ D) $1$

Question 3

The state in Question 2 is conventionally written:

A) $|+\rangle$ B) $|-\rangle$ C) $|i\rangle$ D) $|1\rangle$

Question 4

In the Bloch sphere parametrization $|\psi\rangle = \cos\frac{\theta}{2}|0\rangle + e^{i\varphi}\sin\frac{\theta}{2}|1\rangle$, the state $|+\rangle$ sits at:

A) $\theta = 0$ B) $\theta = \pi/2,\ \varphi = 0$ C) $\theta = \pi/2,\ \varphi = \pi/2$ D) $\theta = \pi$

Question 5

Why does the Bloch parametrization use $\theta/2$ rather than $\theta$?

A) To make the math simpler B) Because orthogonal quantum states ($|0\rangle$, $|1\rangle$) must sit at antipodal points, 180° apart C) Because $\theta$ would exceed $2\pi$ D) It is an arbitrary convention with no physical content

Question 6

Multiplying an entire state by $e^{i\gamma}$ (a global phase):

A) Rotates the Bloch vector by $\gamma$ B) Has no observable consequence whatsoever C) Changes measurement probabilities in the $X$ basis only D) Makes the state unnormalized

Question 7

The relative phase between $|0\rangle$ and $|1\rangle$ components:

A) Is also unobservable B) Is observable — it determines the $\varphi$ coordinate and affects measurements in bases other than $Z$ C) Only matters for two or more qubits D) Is always zero for a physical qubit

Question 8

Measuring $|+\rangle$ in the computational ($Z$) basis yields:

A) $0$ always B) $1$ always C) $0$ or $1$ with equal probability D) $|+\rangle$ again

Question 9

Measuring $|+\rangle$ in the $X$ basis yields:

A) $+$ with certainty B) $-$ with certainty C) $\pm$ equally likely D) Undefined

Question 10

A classical probabilistic bit that is $0$ with probability ½ and $1$ with probability ½ differs from $|+\rangle$ because:

A) It does not — they are the same thing B) $|+\rangle$ has a definite value in the $X$ basis; the probabilistic bit does not C) The probabilistic bit cannot be measured D) $|+\rangle$ is not normalized

Question 11

How many real parameters specify a pure single-qubit state, after removing global phase and enforcing normalization?

A) 1 B) 2 C) 3 D) 4

Question 12

A point strictly inside the Bloch sphere represents:

A) An unnormalized state B) A mixed state C) An entangled state D) An impossible state

Question 13

The Bloch vector of $|1\rangle$ points:

A) $+z$ B) $-z$ C) $+x$ D) $-x$

Question 14

Applying $H$ to $|0\rangle$ produces:

A) $|1\rangle$ B) $|+\rangle$ C) $|-\rangle$ D) $|0\rangle$

Question 15

$T|+\rangle$ equals:

A) $\frac{1}{\sqrt2}(|0\rangle + e^{i\pi/4}|1\rangle)$ B) $\frac{1}{\sqrt2}(|0\rangle + |1\rangle)$ C) $|+\rangle$ rotated by $\pi/4$ about $x$ D) $|1\rangle$

Question 16

True or false: A qubit "stores infinitely much information" because $\alpha$ and $\beta$ are continuous.

Question 17

True or false: Two states differing only by global phase are physically indistinguishable by any measurement.

Question 18

True or false: The Bloch sphere picture extends straightforwardly to two qubits as a pair of spheres.

Question 19

Short answer. A colleague says "a qubit is a bit that's 0 and 1 at the same time." Give a one-sentence correction that a non-physicist can follow.

Question 20

Short answer. You are handed a device that always outputs $0$ and $1$ with 50/50 frequency in the $Z$ basis. Describe one experiment that distinguishes $|+\rangle$ from a classical coin flip.

Answer Key