Self-Assessment Quiz: The Mathematics of Quantum Computing

Twenty questions on bra-ket notation, inner and outer products, unitarity, Hermitian operators, eigenvalues, and tensor products. Answer each before opening the key. Aim for 16+.

Question 1

$\langle\psi|$ is obtained from $|\psi\rangle$ by:

A) Transposing B) Complex conjugating C) Conjugate transposing D) Negating

Question 2

For $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$, the inner product $\langle\psi|\psi\rangle$ equals:

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

Question 3

$\langle\phi|\psi\rangle$ and $\langle\psi|\phi\rangle$ are related by:

A) Equality B) Complex conjugation C) Negation D) No general relation

Question 4

$|0\rangle\langle 0|$ is:

A) A scalar B) A $2\times2$ projection matrix C) A bra D) Undefined

Question 5

An operator $U$ is unitary iff:

A) $U = U^\dagger$ B) $U^\dagger U = I$ C) $U^2 = I$ D) $\det U = 1$

Question 6

Unitarity guarantees, physically:

A) Measurement outcomes are deterministic B) Normalization (total probability) is preserved, and evolution is reversible C) The state stays real-valued D) Gates commute

Question 7

An operator $A$ is Hermitian iff $A = A^\dagger$. Hermitian operators represent:

A) Gates B) Observables C) States D) Measurement outcomes only

Question 8

Eigenvalues of a Hermitian operator are always:

A) Real B) Positive C) On the unit circle D) Complex

Question 9

Eigenvalues of a unitary operator always have:

A) Modulus 1 B) Real values C) Positive values D) Modulus $\le 1$

Question 10

The Pauli $X$ has eigenvalues and eigenvectors:

A) $\pm 1$ with $|0\rangle, |1\rangle$ B) $\pm 1$ with $|+\rangle, |-\rangle$ C) $0, 1$ with $|+\rangle, |-\rangle$ D) $\pm i$ with $|0\rangle, |1\rangle$

Question 11

$|0\rangle \otimes |1\rangle$ written as a column vector is:

A) $(1,0,0,0)^T$ B) $(0,1,0,0)^T$ C) $(0,0,1,0)^T$ D) $(0,0,0,1)^T$

Question 12

For $n$ qubits, the state space has dimension:

A) $2n$ B) $n^2$ C) $2^n$ D) $n!$

Question 13

If $A$ is $2\times2$ and $B$ is $2\times2$, then $A \otimes B$ is:

A) $2\times2$ B) $4\times4$ C) $4\times2$ D) $16\times16$

Question 14

$(A\otimes B)(|\psi\rangle \otimes |\phi\rangle)$ equals:

A) $A|\psi\rangle \otimes B|\phi\rangle$ B) $B|\psi\rangle \otimes A|\phi\rangle$ C) $AB|\psi\rangle\phi\rangle$ D) It does not factor

Question 15

The trace of an operator is:

A) Basis-dependent B) Basis-independent C) Always 1 D) Always 0

Question 16

True or false: Every unitary matrix is Hermitian.

Question 17

True or false: $H$ is both unitary and Hermitian.

Question 18

True or false: Every two-qubit state can be written as a tensor product of two single-qubit states.

Question 19

Short answer. Explain why gate operations must be unitary but measurement is not.

Question 20

Short answer. Given $\langle\phi|\psi\rangle = 0$, what does that tell you physically, and what does it guarantee about measuring $|\psi\rangle$ in a basis containing $|\phi\rangle$?

Answer Key

Q Ans Note
1 C The dagger: transpose and conjugate. Conjugation alone is not enough for complex amplitudes.
2 B The inner product of a state with itself is the sum of squared moduli — equal to 1 for normalized states.
3 B $\langle\phi|\psi\rangle = \overline{\langle\psi|\phi\rangle}$. Inner products are conjugate-symmetric, not symmetric.
4 B An outer product of a ket and a bra yields a matrix; $|0\rangle\langle 0| = \begin{pmatrix}1&0\\0&0\end{pmatrix}$, the projector onto $|0\rangle$.
5 B Equivalently $U^{-1} = U^\dagger$. Note $\det U = 1$ (option D) defines special unitary, a stricter condition.
6 B $\langle\psi|U^\dagger U|\psi\rangle = \langle\psi|\psi\rangle$: lengths preserved, so probabilities still sum to 1, and $U^\dagger$ undoes $U$.
7 B Observables must have real eigenvalues (measurement outcomes are real numbers), which Hermiticity guarantees.
8 A The defining consequence of Hermiticity.
9 A Unitaries preserve length, so eigenvalues are pure phases $e^{i\lambda}$. This is what phase estimation (Ch. 16) extracts.
10 B $X|+\rangle = +|+\rangle$, $X|-\rangle = -|-\rangle$.
11 B $(1,0)\otimes(0,1) = (0,1,0,0)$: index 1 in binary is $01$.
12 C Exponential in qubit count — the fact that makes classical simulation hard.
13 B Dimensions multiply: $(2\cdot2)\times(2\cdot2)$.
14 A Tensor products act factor-wise. This is why independent operations on separate qubits compose cleanly.
15 B Trace is invariant under similarity transformation, hence basis-independent — which is why it appears in physically meaningful quantities like purity $\mathrm{Tr}(\rho^2)$.
16 False $S = \mathrm{diag}(1, i)$ is unitary but not Hermitian ($S^\dagger = \mathrm{diag}(1,-i) \ne S$).
17 True $H = H^\dagger$ and $H^2 = I$, so $H$ is its own inverse — which is why applying $H$ twice returns the original state.
18 False Entangled states, e.g. $(|00\rangle + |11\rangle)/\sqrt2$, are precisely those that do not factor. Non-factorizability is the definition of entanglement.
19 — Gates model closed-system evolution, which is reversible and probability-preserving — exactly unitarity. Measurement is an interaction with the environment: it is irreversible and projects the state onto an eigenspace, so it cannot be unitary. The projectors $P_i = |i\rangle\langle i|$ are Hermitian, not unitary.
20 — Orthogonality means the states are perfectly distinguishable. Measuring $|\psi\rangle$ in a basis containing $|\phi\rangle$ yields outcome $\phi$ with probability $|\langle\phi|\psi\rangle|^2 = 0$ — never. This is the basis of unambiguous state discrimination and of why $|0\rangle$ and $|1\rangle$ can encode a classical bit reliably.