Self-Assessment Quiz: Quantum Error-Correcting Codes

Twenty questions on the stabilizer formalism, the Shor and Steane codes, code parameters, and logical operators. Aim for 16+.

Question 1

A stabilizer code is defined by:

A) An abelian subgroup of the Pauli group not containing $-I$ B) A single unitary C) A classical code only D) A measurement basis

Question 2

For an $[[n,k,d]]$ stabilizer code, the number of independent generators is:

A) $n$ B) $k$ C) $n-k$ D) $d$

Question 3

The code space is:

A) The $+1$ eigenspace of all stabilizer generators B) The null space of the Hamiltonian C) All $2^n$ states D) A single state

Question 4

Stabilizer generators must:

A) Commute pairwise B) Anticommute C) Be unitary but not Hermitian D) Be classical

Question 5

The reason generators must commute is:

A) They must be simultaneously measurable B) To reduce qubit count C) To increase distance D) Convention only

Question 6

Logical operators are Pauli operators that:

A) Commute with all stabilizers but are not themselves in the stabilizer group B) Anticommute with all stabilizers C) Are in the stabilizer group D) Are non-Pauli

Question 7

The code distance $d$ is:

A) The minimum weight of a logical operator B) The number of stabilizers C) $n - k$ D) The number of qubits

Question 8

The Shor code has parameters:

A) $[[9,1,3]]$ B) $[[7,1,3]]$ C) $[[5,1,3]]$ D) $[[9,3,3]]$

Question 9

The smallest code correcting any single-qubit error uses:

A) 3 qubits B) 5 qubits C) 7 qubits D) 9 qubits

Question 10

The quantum Hamming bound for a code correcting $t$ errors requires:

A) $2^{n-k} \ge \sum_{j=0}^{t}\binom{n}{j}3^j$ B) $n \ge 3t$ C) $k \ge t$ D) $d = n$

Question 11

The factor $3^j$ appears because:

A) Each erred qubit can suffer $X$, $Y$, or $Z$ B) There are 3 qubits C) The code rate is 1/3 D) Convention

Question 12

A degenerate code is one where:

A) Distinct errors can produce the same syndrome yet act identically on the code space B) The code fails C) $k=0$ D) Distance is 1

Question 13

Transversal gates are desirable because:

A) They do not spread errors within a code block B) They are faster C) They need fewer qubits D) They are classical

Question 14

The Eastin–Knill theorem states:

A) No code has a transversal universal gate set B) All gates are transversal C) Codes cannot correct errors D) Thresholds do not exist

Question 15

Syndrome extraction requires ancillas because:

A) Measuring data qubits directly would collapse the logical state B) Ancillas are cheaper C) It is faster D) Stabilizers are non-Hermitian

Question 16

True or false: A stabilizer code's logical operators are unique.

Question 17

True or false: Larger distance always means better performance in practice.

Question 18

True or false: Every quantum code is a CSS code.

Question 19

Short answer. Explain why the code space is $2^k$-dimensional given $n-k$ stabilizer generators.

Question 20

Short answer. Why does the $[[5,1,3]]$ code achieve with 5 qubits what Shor needs 9 for?

Answer Key

Q Ans Note
1 A Abelian so generators are simultaneously measurable; excluding $-I$ so the code space is non-trivial.
2 C Each independent generator halves the space.
3 A The simultaneous $+1$ eigenspace.
4 A Abelian by definition.
5 A Non-commuting observables cannot be measured together, so syndrome extraction would be impossible.
6 A They preserve the code space (commute with stabilizers) but act non-trivially on encoded information.
7 A The lightest error that goes undetected and corrupts the logical state.
8 A Nine qubits: three-qubit phase code concatenated with three-qubit bit code.
9 B The perfect $[[5,1,3]]$ code, provably minimal.
10 A The quantum sphere-packing bound.
11 A Three non-identity Paulis per erred qubit — the quantum-specific factor.
12 A Degeneracy lets a code beat naive counting bounds; the surface code is highly degenerate.
13 A A bitwise gate cannot propagate a single-qubit error into a multi-qubit error inside one block — the essence of fault tolerance.
14 A Which forces magic-state distillation for non-Clifford gates (Ch. 9).
15 A Stabilizer eigenvalues are copied onto ancillas, which are measured instead.
16 False Logical operators are defined up to multiplication by stabilizers, so there are many equivalent representatives. Choosing a minimum-weight one gives the distance.
17 False Larger distance costs more qubits and more syndrome-extraction circuitry, and each extra round can introduce errors. Beyond the point where logical error is dominated by other factors, increasing $d$ wastes resources — there is an optimal distance for a given physical error rate and target logical error.
18 False CSS codes are the subclass built from two classical codes with separated $X$ and $Z$ stabilizers. The $[[5,1,3]]$ code is a stabilizer code but not CSS — its generators mix $X$ and $Z$ on the same qubits.
19 — Each independent stabilizer generator is a Hermitian operator with eigenvalues $\pm1$, and requiring $+1$ halves the dimension of the space. Starting from $2^n$ and imposing $n-k$ independent constraints gives $2^n/2^{n-k} = 2^k$ — exactly $k$ logical qubits' worth of space.
20 — Shor's code is built by concatenation: a three-qubit phase-flip code whose every qubit is itself a three-qubit bit-flip code, giving $3\times3=9$ and treating the two error types separately and somewhat wastefully. The $[[5,1,3]]$ code instead uses stabilizers that mix $X$ and $Z$ on the same qubits, protecting against both error types simultaneously rather than in separate layers. Counting confirms 5 is minimal: correcting one error on $n$ qubits requires $2^{n-k} \ge 1 + 3n$, so for $k=1$, $2^{n-1}\ge 1+3n$, which first holds at $n=5$.