Self-Assessment Quiz: Qubit Manipulation in Code

Twenty questions on gates, states, and — above all — phase. Aim for 16 or more.

Question 1

A single-qubit state is described by: - A. one real probability - B. two complex amplitudes with $|\alpha|^2 + |\beta|^2 = 1$ - C. two real probabilities summing to 1 - D. a single complex number

Question 2

Applying a gate to a state is mathematically: - A. matrix–matrix multiplication - B. matrix–vector multiplication - C. a dot product returning a scalar - D. element-wise multiplication

Question 3

Which matrix is H? - A. $\begin{pmatrix}0&1\\1&0\end{pmatrix}$ - B. $\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ - C. $\frac{1}{\sqrt2}\begin{pmatrix}1&1\\1&-1\end{pmatrix}$ - D. $\begin{pmatrix}1&0\\0&i\end{pmatrix}$

Question 4

Applying Z to $|0\rangle$ produces: - A. $|1\rangle$ - B. $|0\rangle$, unchanged - C. $|+\rangle$ - D. $-|1\rangle$

Question 5

Why does Z leave $|0\rangle$ unchanged? - A. Z is the identity - B. $|0\rangle$ lies on Z's rotation axis, and a rotation cannot move a point on its own axis - C. Z only acts on superpositions - D. It is a special case in Qiskit's implementation

Question 6

$H|1\rangle$ equals: - A. $|0\rangle$ - B. $(|0\rangle + |1\rangle)/\sqrt2$ - C. $(|0\rangle - |1\rangle)/\sqrt2$ - D. $|1\rangle$

Question 7

$|+\rangle$ and $|-\rangle$ have measurement probabilities that are: - A. identical: 0.5 and 0.5 - B. 1.0 and 0.0 respectively - C. complex - D. different by a factor of $\sqrt2$

Question 8

Statevector(qc): - A. samples the circuit 1,024 times - B. simulates the circuit exactly, with no sampling - C. runs on hardware - D. returns only probabilities

Question 9

Statevector cannot be used on real hardware because: - A. it is too slow - B. no physical process reveals a quantum state; measurement collapses it - C. IBM does not allow it - D. hardware lacks the memory

Question 10

Statevector simulation requires memory proportional to: - A. $n$ - B. $n^2$ - C. $2^n$ - D. constant

Question 11

On the Bloch sphere, $|+\rangle$ sits at: - A. the north pole - B. the south pole - C. $(+1, 0, 0)$ on the $x$ axis - D. the center

Question 12

The Bloch sphere picture fails for: - A. superposition states - B. entangled multi-qubit states - C. the $|1\rangle$ state - D. states with complex amplitudes

Question 13

Running H H on $|0\rangle$ and measuring gives: - A. 50/50 between 0 and 1 - B. 0 with certainty - C. 1 with certainty - D. a random result each run

Question 14

Running H Z H on $|0\rangle$ and measuring gives: - A. 50/50 - B. 0 with certainty - C. 1 with certainty - D. it depends on the shot count

Question 15

That difference is possible because: - A. Z changes the measurement probabilities - B. Z changes a relative phase, and the second H converts phase into amplitude via interference - C. the simulator is buggy - D. Z is not really a gate

Question 16

$HZH$ equals which gate? - A. I - B. X - C. Y - D. H

Question 17

For $R_y(\theta)$ applied to $|0\rangle$, $P(0)$ equals: - A. $\cos(\theta)$ - B. $\cos^2(\theta)$ - C. $\cos^2(\theta/2)$ - D. $\theta/\pi$

Question 18

Global phase is: - A. observable in any measurement - B. unobservable — but becomes observable when the operation is controlled - C. always zero - D. the same thing as relative phase

Question 19

To compare two states that may differ by a global phase, use: - A. == - B. .equiv() - C. is - D. numpy.array_equal

Question 20

On current superconducting hardware, rz is: - A. the most expensive single-qubit gate - B. virtual — zero duration, zero error - C. not supported - D. implemented as three sx pulses

Answers

# Answer Why
1 B Two complex amplitudes, normalized. §3.2
2 B A $2\times2$ matrix times a length-2 vector. §3.8
3 C Note the minus sign — it is what makes $H\lvert1\rangle$ differ from $H\lvert0\rangle$. §3.2
4 B Unchanged. §3.2
5 B Geometry, not a special case. §3.4
6 C The minus sign is the whole story. §3.2
7 A Identical — which is exactly why §3.5 is surprising. §3.2
8 B Exact, no shots, no noise. §3.3
9 B Measurement collapses the state. §3.3, and Ch. 1 §1.4
10 C The exponential wall again. §3.3
11 C $\lvert+\rangle$ and $\lvert-\rangle$ are the $x$ poles. §3.4
12 B Entangled qubits have no individual states. §3.4 pitfall
13 B The $\lvert1\rangle$ amplitudes cancel. §3.5
14 C The $\lvert0\rangle$ amplitudes cancel instead. §3.5
15 B Interference. This is the chapter's central idea. §3.5
16 B Verified in three lines with Operator. §3.8
17 C The half angle. A persistent source of factor-of-two bugs. §3.6
18 B The exception is what makes phase kickback work. §3.7
19 B == fails on a global phase and the failure is confusing. §3.7
20 B Implemented as a phase-reference shift in the controller. §3.8

Topic Map

Questions Topic Section If you missed these
1, 2, 3, 6 Gates as matrices §3.2, §3.8 Do Exercise 3.2 from memory, then check with Operator
4, 5, 11, 12 The Bloch sphere §3.4 Run Exercise 3.9; the geometry makes the algebra unnecessary
7, 13, 14, 15, 16 Phase and interference §3.5 Reread §3.5 and run code/example-03-phase-interference.py. This is the chapter
8, 9, 10 Statevector §3.3 You will use this tool in every remaining chapter
17 Rotations §3.6 The half-angle convention; Exercise 3.13
18, 19 Global vs. relative phase §3.7 Do Exercise 3.14 — the controlled $-I$ result is worth seeing
20 Virtual $Z$ §3.8 Exercise 3.15; it shapes how you write circuits from Chapter 28 on

Score 16+: go to Chapter 4.

Score 12–15: if you lost points on 13–16, that is the cluster to fix before continuing — Part IV is entirely built on interference and nothing there will make sense without it. Run code/example-03-phase-interference.py and work through the 📐 Math Aside in §3.5 with a pencil.

Score under 12: reread §3.2 through §3.5 and run every snippet. Skip §3.6 to §3.8 on this pass; they are reference material you can return to. The single thing to understand before Chapter 4 is why H Z H gives a different answer from H H.