Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 342 / 2 / a / Solution

The commuting termsare independent stabilizer generators. Since $J>0$, every ground state has $S_j=+1$. The resulting two-dimensional stabilizer subspace isthe phase-flip repetition code. A convenient pair of logical Pauli operators isIndeed, they commute with every stabilizer, anticommute with each other, and are not stabilizers.

For phase-flip errors $E_A={\prod }_{j\in A}{Z}_j$ and $E_B={\prod }_{j\in B}{Z}_j$, the operator entering the Knill--Laflamme condition is $E_A^{†}{E}_B=E_{A△B}$ and has weight at most $2t$. Every nonempty proper product of $Z_j$ anticommutes with some $S_j$ unless it is the full logical operator $\overline{X}$. The Knill--Laflamme conditions therefore hold whenever $2t<n$, and fail once two allowed errors can differ by $\overline{X}$. Thus

Each physical $X_j$ commutes with all stabilizers, and $X_j{X}_1$ is a product of stabilizers. Hence every $X_j$ acts on the code as the undetectable logical operator $\overline{Z}$. The code cannot detect, and therefore cannot correct, even a single bit flip.