Questions: Stabilizer Codes

Question 1

In an [[n,k,d]] stabilizer code, the stabilizer group S has n-k independent generators. What do the remaining k degrees of freedom correspond to?

• A. k physical qubits that are not used for error correction
• B. k logical qubits whose information is encoded in the code space
• C. k possible error syndromes
• D. k ancilla qubits needed for syndrome measurement

Answer: (B)

The n-k stabilizer generators constrain the state to lie in a 2^k-dimensional code subspace of the 2^n-dimensional Hilbert space. This 2^k-dimensional space encodes k logical qubits. The logical operators (logical X and Z for each logical qubit) commute with all stabilizers but are not themselves in the stabilizer group — they act within the code space to manipulate the encoded information.

Question 2

If two different errors E1 and E2 produce the same syndrome, the stabilizer code cannot distinguish them and error correction fails.

• T. True
• F. False

Answer: False

Two errors with the same syndrome differ by an element of the stabilizer group: E1*E2^dagger is in the stabilizer (or is proportional to a logical operator). If E1*E2^dagger is in the stabilizer, both errors have the same effect on the code space, so correcting either one corrects both — no failure occurs. Error correction fails only when E1*E2^dagger is a nontrivial logical operator, meaning the two errors differ by an undetectable logical operation. This is why the minimum distance d matters: it determines the weight of the lightest undetectable error.

Question 3

How do CSS codes simplify the construction of quantum error-correcting codes?

Model answer: CSS (Calderbank-Shor-Steane) codes are built from two classical linear codes C1 and C2 with C2 subset of C1. X errors are corrected using C1 and Z errors are corrected using C2^perp, with the two correction procedures operating independently. This means you can design the quantum code by choosing classical codes with the right properties, leveraging decades of classical coding theory rather than designing quantum codes from scratch.

The separation of X and Z error correction in CSS codes is powerful because it reduces quantum code design to classical code design. The Steane code ([[7,1,3]]) is a CSS code built from the classical Hamming [7,4,3] code. The constraint that C2 is contained in C1 ensures that X and Z corrections do not interfere with each other. CSS codes are also the foundation for surface codes, the leading candidates for practical quantum error correction.