Editing Quantum error correction (section)

==Sign-flip code==
[[File:Quantum error correction of phase flip using three qubits.svg|upright=1.5|thumb|right|[[Quantum circuit]] of the phase-flip code]]
The bit flip is the only kind of error in classical computers. In quantum computers, however, another kind of error is possible: the sign flip. Through transmission in a channel, the relative sign between <math>|0\rangle</math> and <math>|1\rangle</math> can become inverted. For instance, a qubit in the state <math>|-\rangle=(|0\rangle-|1\rangle)/\sqrt{2}</math> may have its sign flip to <math>|+\rangle=(|0\rangle+|1\rangle)/\sqrt{2}.</math>

The original state of the qubit
<math display="block">|\psi\rangle = \alpha_0|0\rangle+\alpha_1|1\rangle</math>
will be changed into the state
<math display="block">|\psi'\rangle = \alpha_0|{+}{+}{+}\rangle+\alpha_1|{-}{-}{-}\rangle.</math>

In the Hadamard basis, bit flips become sign flips and sign flips become bit flips. Let <math>E_\text{phase}</math> be a quantum channel that can cause at most one phase flip. Then the bit-flip code from above can recover <math>|\psi\rangle</math> by transforming into the Hadamard basis before and after transmission through <math>E_\text{phase}</math>.