Checkpoint: Analyze a Code

This checkpoint pulls the module together: given only a code's stabilizer generators, you will read off its parameters and compute its distance directly — the core analytical skill of error-correction theory.

The code under study: $[[4,2,2]]$

The $[[4,2,2]]$ code is the smallest CSS code and the smallest code that can detect an arbitrary single-qubit error. It has just two stabilizer generators on four qubits,

which commute because they overlap on all four qubits (an even number). With $n=4$ physical qubits and $n - k = 2$ generators, it encodes

Reading off logical operators

We need $2k = 4$ logical operators obeying the qubit algebra and commuting with both generators. A standard choice is

each of weight 2. Verify that every one commutes with $XXXX$ and $ZZZZ$ (even overlaps) and that the intended pairs anticommute — $\bar X_1$ and $\bar Z_1$ share only qubit 1, $\bar X_2$ and $\bar Z_2$ share only qubit 1 — while different-index pairs commute (each shares an even number of qubits) — exactly the constraints from the logical-operators lesson.

Computing the distance

The distance is the minimum weight of any operator in $N(\mathcal S)\setminus\mathcal S$. The lightest logical operators above have weight 2, and no weight-1 Pauli both commutes with $XXXX$ and $ZZZZ$ and lies outside $\mathcal S$ — so

The code is therefore $[[4,2,2]]$: it detects any single error (distance 2 means $d-1 = 1$ detectable) but corrects $\lfloor (d-1)/2\rfloor = 0$ of them. It is a detection code, the entry point to the CSS and surface-code constructions of the next module.

Try it

Compute the distance of the $[[4,2,2]]$ code from its two generators $XXXX$ and $ZZZZ$ using the symplectic formalism: enumerate the Paulis in $N(\mathcal S)\setminus\mathcal S$ and return the minimum weight. You should obtain $d = 2$.

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