Five-qubit error correcting code

The five-qubit error correcting code or the [[5,1,3]] code, is the smallest quantum error correcting code that can protect a logical qubit from any arbitrary single qubit error.^[1] In this code, 5 physical qubits are used to encode the logical qubit.^[2] With ${\displaystyle X}$ and ${\displaystyle Z}$ being Pauli matrices and ${\displaystyle I}$ the Identity matrix, this code's generators are ${\displaystyle \langle XZZXI,IXZZX,XIXZZ,ZXIXZ\rangle }$. Its logical operators are ${\displaystyle {\bar {X}}=XXXXX}$ and ${\displaystyle {\bar {Z}}=ZZZZZ}$.^[3] Once the logical qubit is encoded, errors on the physical qubits can be detected via stabilizer measurements. A lookup table that maps the results of the stabilizer measurements to the types and locations of the errors gives the control system of the quantum computer enough information to correct errors.^[4]