Shor's Code: Understanding how it satisfies Knill Laflamme Theorem

Question

I'm new to Quantum Error Correction, and I have a question on Shor's Code.

If we have a protected subspace, $V \subset \mathbf{C}^2\otimes \cdots \otimes \mathbf{C}^2$

$V=\operatorname{span}\{|0_{l}\rangle, |1_{L}\rangle.$ We also consider Pauli basis of $\mathbf{C}^2\otimes \cdots \otimes \mathbf{C}^2$ of 9 copies, and constructed as follows: Take the basis of $M_2$ consisting of: \begin{eqnarray} \nonumber X=\begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}, Y= \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}, Z=\begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix} & \text{and} & 1_2. \end{eqnarray} We list the 1-Paulis as $U_1,\cdots ,U_{28}.$ Define the error map as $\mathscr{E}:M_{2^9}\rightarrow M_{2^9}$ by $\mathscr{E}(X)=\frac{1}{28}\sum_{i=1}^{28}U_iXU_i^*$. $\mathscr{E}$ is completely positive and trace preserving. How do we say that it satisfies the Knill Laflamme Theorem and thus ensure the existence of a recovery operator?

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Cross-posted on math.SE

Answer 1

Strictly, what you have to calculate is that for all $i$ and $j$ $$ \langle 0_L|U_iU_j|1_L\rangle=0 $$ and $$ \langle 0_L|U_iU_j|0_L\rangle=\langle 1_L|U_iU_j|1_L\rangle. $$ (I've ignored the Hermitian conjugate because all the single-qubit errors are Hermitian.)

Obviously there's a lot of work involved in calculating all $28^2$ cases of $i,j$. You can at least simplify this by using symmetry - there's permutation invariance within blocks of three qubits and between blocks of three qubits. This means that you can reduce your work to two sets of cases: (i) two Pauli errors on the same block of 3 qubits (of which there are $9^2$ cases, but we can take the two errors to be on the first two qubits, or two on the first, reducing to $3+3^2$ cases) and (ii) one Pauli error on each of 3 qubits (of which there are $9^2$ cases, but we can assume the errors are on the first qubit of each block, reducing to $3^2$ cases). 21 error combinations is a much more tolerable calculation.

Could I also recommend finding out about the stabilizer formalism as applied to error correcting codes? This places the calculation in a completely different light which is much more convenient.