Quantum Error Correction Code

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 $\langle i{C}|D{a}| j{C}\rangle \neq C{a} \delta_{i j}$.
• A code with distance can correct errors.

K-L Criterion

The necessary and sufficient condition for a code:
$
\left\langle j{C}\left|E{b}^{\dagger} E{a}\right| i{C}\right\rangle=C{a b} \delta{i j}
Ea, E_bC{ab}$ is an arbitary complex matrix.

If $C{ab}\neq \delta{ab}$, the code is called degenerate code, otherwise the code is non-degenerate code. The 9-bit Shor code is a degenerate code.

Stabilizer formalism

• 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.

AQEC(Approximate quantum error correction code)

Entanglement Fidelity

worst-case entanglement fidelity: input-independent.