t b a b a

Similarly, a qubit in state |1〉 will change into a state:
|1〉|E〉 ↣ β3|1E3〉 + β4|0E4〉
Therefore, an arbitrary state |ψ〉 = α0|0〉 + α1|1〉 interacts with the environment |E〉 as:
(α0|0〉 + α1|1〉)|E〉 ↣ α0β1|0E1〉 + α0β2|1E2〉 + α1β3|1E3〉 + α1β4|0E4〉
= 1
2 (α0|0〉 + α1|1〉)(β1|E1〉 + β3|E3〉)
+ 1
2 (α0|0〉 − α1|1〉)(β1|E1〉 − β3|E3〉)
+ 1
2 (α0|1〉 + α1|0〉)(β2|E2〉 + β4|E4〉)
+ 1
2 (α0|1〉 − α1|0〉)(β2|E2〉 − β4|E4〉)
= 1
2 |ψ〉(β1|E1〉 + β3|E3〉) + 1
2 Z|ψ〉(β1|E1〉 − β3|E3〉)
+ 1
2 X|ψ〉(β2|E2〉 + β4|E4〉) + 1
2 XZ|ψ〉(β2|E2〉 − β4|E4〉)
This represents the most general evolution that may occur on a single qubit, and the last
3 terms represent phase-flip, bit-flip and bit-phase-flip respectively.
Similar to classical error correction, quantum error correction also operates by the ju-
dicious use of redundancy, that is, making the system larger than the information itself,
which could make it more resistent to perturbations. As we discussed in section 2.6, many
different properties from classical error correction must be carefully considered when we
construct quantum error-correcting codes. The first quantum error correcting codes were
67