t b a b a

Quantum error correction is used in quantum computing to protect quantum information
from errors due to decoherence and other quantum noise. It is essential for fault-tolerant
quantum computation. It is designed to deal not just with noise during quantum information
communication, but also with the protection of stored information. In this case, the user en-
codes the information in a storage system and retrieves it at a later time. Since the quantum
information is not absolutely isolated from the environment(actually the influence is much
larger than in the classical world), any error-correction method applicable to communication
is also applicable to storage and vice versa. Furthermore, quantum error correction is also
essential for faulty quantum gates, faulty quantum preparation, and faulty measurements.
The discovery of powerful error correction methods therefore caused much excitement, since
it converted large scale quantum computation from a practical impossibility to a possibility.
An error is a unitary transformation of the state space. The space of errors for a single
qubit is spanned by the four Pauli matrices as shown in Table 5.1.
Table 5.1: Quantum errors for a single qubit are spanned by the Pauli matrices
I no error | 0〉 →| 0〉 | 1〉 →| 1〉
X bit error | 0〉 →| 1〉 | 1〉 →| 0〉
Z phase error | 0〉 →| 0〉 | 1〉 → − | 1〉
Y = iXZ combination | 0〉 → i | 1〉 | 1〉 → −i | 0〉
Generally, the evolution of a qubit in state |0〉 interacting with an environment |E〉 will
yield a state as:
|0〉|E〉 ↣ β1|0E1〉 + β2|1E2〉
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