The Exercise

The Shor code encodes one qubit in nine, grouped in three blocks of three. Exercise 10.10 is to verify that the code satisfies the quantum error correction conditions for 28 errors: the identity $I$, and the Pauli operators $X_k$, $Y_k$ and $Z_k$ that act on a single qubit $k$ of the nine and as the identity on the rest, for $k = 1, \ldots, 9$.^✦

The library states this for the projector $P$ onto the code space and the family of the 28 error operators acting on the nine qubits.

Qubits and Registers

A quantum system $S$ is given by its state space, a finite dimensional complex Hilbert space.^✦ The composite system $S \otimes T$ of two quantum systems $S$ and $T$ is the quantum system whose state space is the tensor product of their state spaces.^✦ The qubit is the quantum system whose state space is $\mathbb{C}^2$.^✦ Three qubits make up the three qubit register^✦, and three copies of that register, one per block, make up the nine qubit Shor register: qubits $1, 2, 3$ form the first block, $4, 5, 6$ the second and $7, 8, 9$ the third.^✦

A pure state $\psi$ of a quantum system $S$ is a vector of unit norm, $\lVert \psi \rVert = 1$, in the state space of $S$. For a nonzero vector $v$ in that space, the normalized state of $v$ is the pure state $v / \lVert v \rVert$.^✦ For pure states $\psi$ of $S$ and $\varphi$ of $T$, the product $\psi \otimes \varphi$ is the pure state of $S \otimes T$ whose vector is the tensor product of the two vectors.^✦

The computational basis states $|0\rangle$ and $|1\rangle$ of the qubit are the pure states whose vectors are the standard basis vectors $e_0$ and $e_1$ of $\mathbb{C}^2$.^✦ On the three qubit register, the basis state $|q_1 q_2 q_3\rangle$ for bits $q_1, q_2, q_3$ is the product $|q_1\rangle \otimes |q_2\rangle \otimes |q_3\rangle$.^✦

The Code

For a bit $s$, the block cat vector is $|000\rangle + (-1)^s |111\rangle$ in the state space of the three qubit register^✦, and the block cat state is its normalized state, $$|\mathrm{cat}_s\rangle = \frac{|000\rangle + (-1)^s |111\rangle}{\sqrt{2}}.$$(1)^✦ Given bits $s_1, s_2, s_3$, the product of three cat blocks is the pure state $|\mathrm{cat}_{s_1}\rangle \otimes |\mathrm{cat}_{s_2}\rangle \otimes |\mathrm{cat}_{s_3}\rangle$ of the nine qubit register. The Shor codewords $|0_L\rangle$ and $|1_L\rangle$ are the products with all three bits $0$ and with all three bits $1$: $$|0_L\rangle = \frac{(|000\rangle + |111\rangle)^{\otimes 3}}{2\sqrt{2}}, \qquad |1_L\rangle = \frac{(|000\rangle - |111\rangle)^{\otimes 3}}{2\sqrt{2}}.$$(2)^✦ The Shor code projector $P$ is the projector onto the code space spanned by the two codewords, $$P = |0_L\rangle\langle 0_L| + |1_L\rangle\langle 1_L|.$$(3)^✦

The Errors

An observable $A$ of a quantum system $S$ is a self adjoint operator on the state space of $S$, $A^\dagger = A$.^✦ The identity observable $I$ of $S$ has the identity as its operator.^✦ For observables $A$ of $S$ and $B$ of $T$, the tensor product $A \otimes B$ is the observable of $S \otimes T$ whose operator is the tensor product of their operators.^✦

The Pauli matrices are $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},$$(4) and for a real vector $\vec{n} = (n_1, n_2, n_3)$ the matrix $\vec{n} \cdot \vec{\sigma}$ is the complex $2 \times 2$ matrix $n_1 X + n_2 Y + n_3 Z$.^✦ The Pauli observable $\vec{n} \cdot \vec{\sigma}$ is the observable of the qubit with this matrix as its operator on $\mathbb{C}^2$.^✦ Along the unit vectors $\hat{n} = (1, 0, 0)$, $(0, 1, 0)$ and $(0, 0, 1)$ it gives the observables $X$, $Y$ and $Z$, the three types of single qubit error, $\sigma_1 = X$, $\sigma_2 = Y$ and $\sigma_3 = Z$.^✦

Placing a qubit observable $A$ on one qubit means letting it act there and as the identity elsewhere. On the three qubit register, $A$ on position $p$ is $A \otimes I \otimes I$, $I \otimes A \otimes I$ or $I \otimes I \otimes A$.^✦ On the nine qubit register, $A$ on position $p$ of block $b$ acts in this way on block $b$ and as the identity on the other two blocks; it is $A_k$, the observable $A$ on qubit $k = 3(b - 1) + p$.^✦ The Pauli error of type $t$ on that qubit is the operator of $\sigma_t$ placed there.^✦

The 28 errors are labelled by $0$, for the identity, and by the 27 triples $(t, b, p)$ of a type, a block and a position, each in $\{1, 2, 3\}$.^✦ The Shor error operators on the nine qubit register are $$E_0 = I, \qquad E_{(t, b, p)} = (\sigma_t)_{3(b - 1) + p}.$$(5)^✦

The Claim

An operator $P$ on the state space of a quantum system $S$ and a family of operators $E_i$ on that space, indexed by a set $\mathcal{I}$, satisfy the quantum error correction conditions when there is a Hermitian complex matrix $\alpha$, with rows and columns indexed by $\mathcal{I}$, such that $P E_i^\dagger E_j P = \alpha_{ij} P$ for all $i$ and $j$ in $\mathcal{I}$.^✦

What Is Checked

The claim is stated over the Shor code projector $P$ of (3) and the 28 error operators $E_i$ of (5).

About these constructions the library proves that the block cat vector $|000\rangle + (-1)^s |111\rangle$ is nonzero for both values of $s$^✦, so that its normalized state exists, and that a product of pure states has unit norm, the norm of a product vector being the product of the norms. It proves also that $\vec{n} \cdot \vec{\sigma}$ is Hermitian and that a tensor product of Hermitian operators is Hermitian, so that each Pauli error placed on one qubit is an observable of the nine qubit register.