The Exercise

A stabilizer code on $n$ qubits is fixed by its stabilizer $S$, a group of Pauli operators that does not contain $-I$; the code is the space of states left unchanged by every element of $S$. Noise strikes the qubits with one of finitely many Pauli errors $E_j$. The question is when a single recovery operation, the same whatever error occurred, restores every state of the code.

Theorem 10.8 answers it with a condition on the products $E_j^{-1} E_k$ of two errors. The condition is phrased through the normalizer $N(S)$ of $S$, the set of Pauli operators $g$ with $g S g^{-1} = S$.

The library states the theorem for the Pauli operators as matrices on the $n$ qubits, with the code given by its projector and the recovery by its operation elements.

The Pauli Group

The Pauli matrices are the four complex $2 \times 2$ matrices $\sigma_0 = I$, $\sigma_1 = X$, $\sigma_2 = Y$ and $\sigma_3 = Z$, with rows and columns indexed by $\{0,1\}$, where $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$(1)^✦

The state space of $n$ qubits has a basis vector $|x\rangle$ for each bit string $x = x_1 \cdots x_n$ in $\{0,1\}^n$, and the entries of an operator on it are indexed by two such strings $x$ and $y$. For a choice $a$ of one Pauli matrix $\sigma_{a_k}$ on each qubit $k$, the Pauli string $\sigma_{a_1} \otimes \cdots \otimes \sigma_{a_n}$ is the operator on the $n$ qubits whose matrix entries are products of single qubit entries, $$\langle x | \, \sigma_{a_1} \otimes \cdots \otimes \sigma_{a_n} \, | y \rangle = \prod_{k=1}^{n} (\sigma_{a_k})_{x_k y_k}.$$(2)^✦

The Pauli group $G_n$ on $n$ qubits consists of the operators $i^s P$, with $P$ a Pauli string and $s$ an integer modulo $4$, the phase power.^✦ The phase $i^s$ is $i$ raised to the representative of $s$ in $\{0,1,2,3\}$, one of the four fourth roots of unity.^✦ An element of $G_n$ acts on the $n$ qubits by the complex $2^n \times 2^n$ matrix $$i^s \, \sigma_{a_1} \otimes \cdots \otimes \sigma_{a_n}.$$(3) The element $-I$ of $G_n$ is the identity string $I \otimes \cdots \otimes I$ with phase power $s = 2$, so that $i^2 I = -I$.^✦

The Code and the Recovery

The code projector $P_S$ of a subgroup $S$ of $G_n$ is its stabilizer sum, the sum of the matrices of all the elements of $S$, divided by the order $|S|$ of $S$: $$P_S = \frac{1}{|S|} \sum_{s \in S} s.$$(4)^✦

A recovery $\mathcal{R}$ on the $n$ qubits is a quantum operation, a map $\rho \mapsto \sum_o R_o \rho R_o^\dagger$ given by finitely many operation elements $R_o$ indexed by $o$, operators on the state space of the qubits, and it is trace preserving when $$\sum_o R_o^\dagger R_o = I.$$(5)

The Claim

What Is Checked