Five-qubit error correcting code ($[[5,1,3]]$)

The five-qubit code encodes one logical qubit into five physical qubits and corrects an arbitrary error on one physical qubit. In a nondegenerate realization, the identity and fifteen one-qubit Pauli operators take the two-dimensional code into sixteen mutually orthogonal syndrome subspaces, filling the 32-dimensional physical Hilbert space.

// Target: quantum-error-correction.bigb

The four commuting independent stabilizer generators define a two-dimensional stabilizer code. The identity and fifteen single-site Pauli errors have sixteen distinct four-bit error syndromes, so their two-dimensional images form an orthogonal direct sum of the physical space. This proves nondegenerate single-site correction and the perfect quantum error-correcting code property. The operators $X^{\otimes 5}$ and $Z^{\otimes 5}$ commute with the stabilizer and anticommute with each other. Multiplying $X^{\otimes 5}$ by $XZZXI$ gives, up to phase, $IYYIX$, a nontrivial weight-three logical operator, proving distance exactly three.