A brief review of QEC code, stabilizer formalism and AQEC code. 记一记我每次看了都忘的东西。未完待续

QEC Notation

: is the number of physical qubits; is the number of logical qubits encoded; id code distance.
Code distance: the minimum weight of Pauli operator such that .
A code with distance can correct errors.

K-L Criterion

The necessary and sufficient condition for a code:

where are errors that the code can correct, is an arbitary complex matrix.

If , the code is called degenerate code, otherwise the code is non-degenerate code. The 9-bit Shor code is a degenerate code.

Stabilizer: If , we say that a state is stabilized by operator .
Motivation: many quantum states can be more easily described by the operators that stabilize them than the state itself.
is the subgroup of n-bits Pauli group , and is the subspace that stabilized by .
For to be non-trivial, must satisfy:(a) is Abelian, i.e. all elements of commute and (b) is not in .
Vector space dimension: Let be generated by independent and commuting elements of , and , then is a -dim vec space.

worst-case entanglement fidelity: input-independent.