Self-Assessment Quiz: Quantum Circuits

Twenty questions on circuit construction, universality, depth, uncomputation, and reading circuit diagrams. Aim for 16+.

Question 1

In a quantum circuit diagram, time flows:

A) Right to left B) Left to right C) Top to bottom D) It is unordered

Question 2

Each horizontal wire in a circuit diagram represents:

A) A classical bit B) A qubit C) A gate D) A measurement

Question 3

Circuit depth is:

A) The total number of gates B) The number of qubits C) The number of time steps, where non-overlapping gates run in parallel D) The number of measurements

Question 4

A circuit with 40 gates on 10 qubits, well parallelized, could have depth as low as:

A) 40 B) 10 C) 4 D) 1

Question 5

Depth matters more than gate count on real hardware because:

A) Gates are free B) Decoherence is a function of elapsed time C) Depth determines qubit count D) Compilers cannot reduce gate count

Question 6

A universal gate set is one that:

A) Contains every gate B) Can approximate any unitary to arbitrary accuracy C) Contains only two-qubit gates D) Is unique

Question 7

Which is not universal?

A) $\{H, T, \text{CNOT}\}$ B) $\{H, S, \text{CNOT}\}$ C) $\{\text{Toffoli}, H\}$ D) Arbitrary single-qubit rotations + CNOT

Question 8

A barrier in a Qiskit circuit:

A) Applies a gate B) Prevents the compiler from optimizing across it C) Measures a qubit D) Resets a qubit

Question 9

Uncomputation means:

A) Deleting a qubit B) Reversing intermediate operations to return ancillas to $|0\rangle$ C) Measuring and discarding D) Applying $X$ to all qubits

Question 10

Uncomputation is necessary because:

A) Ancillas are expensive B) Entangled "garbage" ancillas destroy interference in the main register C) The compiler requires it D) It reduces depth

Question 11

Applying $U$, then a controlled operation, then $U^\dagger$ is a common pattern called:

A) Conjugation by $U$ B) Uncomputation C) Teleportation D) Reset

Question 12

Measuring an ancilla that is entangled with your data register:

A) Has no effect on the data B) Effectively decoheres the data in that basis C) Is always safe D) Restores coherence

Question 13

A controlled-$U$ where $U = e^{i\alpha}V$ (global phase $\alpha$):

A) Is identical to controlled-$V$ B) Differs from controlled-$V$ — the global phase becomes relative C) Is not unitary D) Is undefined

Question 14

Two gates acting on disjoint qubits:

A) Must be applied in a fixed order B) Commute, and can be applied in the same time step C) Cannot both appear in one circuit D) Always entangle

Question 15

Circuit width refers to:

A) Depth B) Number of qubits C) Number of gates D) Runtime

Question 16

True or false: Every quantum circuit can be reversed by applying the inverse gates in reverse order.

Question 17

True or false: Adding more qubits always increases circuit depth.

Question 18

True or false: A circuit's classical simulation cost is determined only by its qubit count.

Question 19

Short answer. Explain why a "garbage" ancilla left entangled with the data register is worse than an ancilla left in a random but unentangled state.

Question 20

Short answer. You have a coherence budget of 100 μs and two-qubit gates take 400 ns. Estimate your depth budget and say what that implies for algorithm design.

Answer Key

Q Ans Note
1 B Standard convention, opposite to matrix-multiplication order — a frequent source of sign and ordering errors when translating diagrams to matrices.
2 B Wires are qubits persisting through time, not signals travelling through space.
3 C Depth counts layers, not gates.
4 C If gates pack 10-wide into layers, 40 gates fit in 4 layers.
5 B Decoherence is a clock, and depth × gate time is the clock reading. A wide shallow circuit beats a narrow deep one with the same gate count.
6 B Approximation to arbitrary accuracy, not exact representation — the Solovay–Kitaev regime.
7 B Clifford only; simulable classically by Gottesman–Knill.
8 B A compiler directive with no physical effect — useful for preserving structure you want to benchmark.
9 B Run the computation backwards to disentangle and return ancillas to a known state.
10 B Entangled garbage acts as a "which-path" record; tracing it out destroys the very interference the algorithm needs.
11 A Conjugation. Used constantly: basis changes, controlled operations, and error-correction gadgets are all of this form.
12 B Measuring one half of an entangled pair collapses the other. If you did not intend that, it is decoherence you inflicted on yourself.
13 B The control branch picks up $e^{i\alpha}$ while the other does not — the global phase becomes observable. This is phase kickback and is the engine of most quantum algorithms.
14 B Disjoint supports commute; this is what allows parallel layers.
15 B Width = qubits, depth = layers. Resource estimates quote both.
16 True Every gate is unitary, hence invertible; $U^\dagger = U_n^\dagger\cdots U_1^\dagger$. Measurement is the exception that breaks reversibility.
17 False Adding qubits can decrease depth by enabling parallelism or by replacing a deep construction with a wider shallow one — the standard space-time trade.
18 False $T$-count matters at least as much (Ch. 6), and entanglement structure matters for tensor-network methods. Qubit count alone is a poor predictor.
19 — An unentangled ancilla, however randomized, factors out of the state: the data register still has a well-defined pure state and its interference is intact. An entangled ancilla means the data register is described by a mixed reduced state; the relative phases between computational branches are no longer well-defined, so the interference step that produces the answer fails. Entanglement with garbage is decoherence by another name.
20 — $100\,\mu s / 400\,\text{ns} = 250$ layers of two-qubit gates, and in practice gate error bites sooner — at $10^{-3}$ error, 250 sequential two-qubit gates already give fidelity $\approx 0.78$. So the usable depth is on the order of 100 layers. Any algorithm needing thousands of sequential entangling layers is out of reach without error correction, which is precisely why NISQ algorithms (Ch. 18–21) are designed to be shallow and variational rather than deep and exact.