The five-qubit code

“The five-qubit code is the shortest possible Quantum error-correction code to correct one error.”

• Quantum code from two binary linear codes;

“The stabilizer is simply generated by cyclic permutations of σx ⊗ σz ⊗ σz ⊗ σx ⊗ I. There are five cyclic permutations of this, but only four produce independent generators. The stabilizer has sixteen elements: the identity, and the 3 × 5 cyclic permutations of σx ⊗ σz ⊗ σz ⊗ σx ⊗ I, σy ⊗ σx ⊗ σx ⊗ σy ⊗ I, and σz ⊗ σy ⊗ σy ⊗ σz ⊗ I. XL is just the tensor product of five σx’s. ZL is the tensor product of the five σz ’s.”

“The five-qubit code is a linear GF(4) code. Therefore, the operation T : σx → σy , σz → σx applied transversally is a valid fault-tolerant operation and performs an encoded version of itself.”

References

• Gottesman1997;