The Exercise

Shor’s code stores one qubit in nine, grouped in three blocks of three. Its codeword $|0_L\rangle$ puts every block in the cat state $|000\rangle + |111\rangle$, its codeword $|1_L\rangle$ puts every block in $|000\rangle - |111\rangle$, and errors are diagnosed by measuring eight syndrome observables $g_1, \dots, g_8$: products of $Z$ on two adjacent qubits of a block, and products of $X$ over two adjacent blocks. The exercise takes the product of $X$ over all nine qubits as $\bar Z$ and the product of $Z$ over all nine as $\bar X$, and asks to show that the two act on the encoded qubit as its Pauli $Z$ and $X$.

The Nine Qubit Register

A quantum system $S$ is a physical system whose state space is a finite dimensional complex Hilbert space^✦, and any such space $H$ is the state space of a system.^✦ The qubit is the system whose state space is $\mathbb{C}^2$^✦; its basis states $|0\rangle$ and $|1\rangle$ are the standard basis vectors of $\mathbb{C}^2$.^✦ The composite system $S \otimes T$ of systems $S$ and $T$ is the system whose state space is the tensor product of their state spaces.^✦

A pure state $\psi$ of $S$ is a unit vector $|\psi\rangle$ in the state space of $S$.^✦ For a nonzero vector $v$ in the state space of $S$, the normalized pure state of $v$ is the pure state of $S$ with vector $v/\lVert v\rVert$.^✦ For pure states $\psi$ of $S$ and $\varphi$ of $T$, the product $\psi \otimes \varphi$ is the pure state of $S \otimes T$ with vector $|\psi\rangle \otimes |\varphi\rangle$.^✦

The three qubit register is the composite of three qubits.^✦ Its computational basis states are the products $|q_1q_2q_3\rangle = |q_1\rangle \otimes |q_2\rangle \otimes |q_3\rangle$ for bits $q_1$, $q_2$ and $q_3$.^✦ The nine qubit Shor register is the composite of three copies of the three qubit register, one for each block, so that its qubits fall into the blocks $q_1q_2q_3$, $q_4q_5q_6$ and $q_7q_8q_9$.^✦

The Codewords

For a bit $\sigma$, the block cat vector is the vector $|000\rangle + (-1)^\sigma|111\rangle$ in the state space of the three qubit register^✦, and the block cat state is the pure state obtained by normalizing it, $(|000\rangle + (-1)^\sigma|111\rangle)/\sqrt{2}$.^✦ For bits $\sigma_1$, $\sigma_2$ and $\sigma_3$, the product of three cat blocks is the pure state of the Shor register that puts block $b$ in the block cat state of the bit $\sigma_b$.^✦ The Shor codeword $|s_L\rangle$, for a bit $s$, is the product of three cat blocks that all carry the bit $s$: $$|s_L\rangle = \frac{1}{2\sqrt{2}}\,\bigl(|000\rangle + (-1)^s|111\rangle\bigr) \otimes \bigl(|000\rangle + (-1)^s|111\rangle\bigr) \otimes \bigl(|000\rangle + (-1)^s|111\rangle\bigr).$$(1)^✦

Observables and Unitaries

An observable $A$ on a system $S$ is a self adjoint operator on the state space of $S$, $A^\dagger = A$.^✦ A pure state $\psi$ of $S$ is an eigenstate of $A$ with eigenvalue $\mu$, a real number, when $A|\psi\rangle = \mu|\psi\rangle$.^✦ The identity observable $I$ on $S$ is the observable whose operator is the identity^✦, and for observables $A$ on $S$ and $B$ on $T$ the tensor product $A \otimes B$ is the observable on $S \otimes T$ whose operator is the tensor product of their operators.^✦

A unitary $U$ of a system $S$ is an operator on the state space of $S$ with $U^\dagger U = UU^\dagger = 1$.^✦ Applied to a pure state $\psi$ of $S$, the unitary $U$ gives the pure state with vector $U|\psi\rangle$.^✦ For unitaries $U$ of $S$ and $V$ of $T$, the product unitary $U \otimes V$ is the unitary of $S \otimes T$ whose operator is the tensor product of their operators.^✦

The Pauli matrices are $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},$$(2) and for a real vector $\vec n = (n_x, n_y, n_z)$ the matrix $\vec n \cdot \vec\sigma$ is $n_x X + n_y Y + n_z Z$.^✦ For such $\vec n$, the Pauli observable $\vec n \cdot \vec\sigma$ is the observable on the qubit whose operator is the matrix $\vec n \cdot \vec\sigma$^✦; the observables $X$ and $Z$ are those for the unit vectors $\hat n = (1, 0, 0)$ and $\hat n = (0, 0, 1)$. The Pauli $Z$ gate, the phase flip, is the unitary of the qubit whose operator is the matrix $Z$.^✦

The Logical Operators and the Syndromes

Write $X_j$ and $Z_j$ for $X$ and $Z$ acting on qubit $j$ of a register and as the identity on its other qubits. On the three qubit register, the block observable $X_1X_2X_3 = X \otimes X \otimes X$ is the observable that acts as $X$ on each qubit^✦, and the block gate $Z_1Z_2Z_3 = Z \otimes Z \otimes Z$ is the unitary that applies the Pauli $Z$ gate to each qubit.^✦ The bit flip code stores a qubit in the three qubit register, and its two syndrome observables are $Z_1Z_2 = Z \otimes Z \otimes I$, which compares the first qubit with the second^✦, and $Z_2Z_3 = I \otimes Z \otimes Z$, which compares the second with the third.^✦

On the Shor register, the logical operator $\bar X$ is the product unitary of three block gates^✦, and the logical operator $\bar Z$ is the tensor product of three block observables: $$\bar X = (Z \otimes Z \otimes Z)^{\otimes 3} = Z_1Z_2 \cdots Z_9, \qquad \bar Z = (X \otimes X \otimes X)^{\otimes 3} = X_1X_2 \cdots X_9.$$(3)^✦ The Shor code has eight syndrome observables $g_1, \dots, g_8$: the two bit flip syndromes in each block, and the block observable $X \otimes X \otimes X$ on each pair of adjacent blocks, with the identity observable on the remaining block: $$\begin{gathered} g_1 = Z_1Z_2, \quad g_2 = Z_2Z_3, \quad g_3 = Z_4Z_5, \quad g_4 = Z_5Z_6, \quad g_5 = Z_7Z_8, \quad g_6 = Z_8Z_9, \\ g_7 = X_1X_2 \cdots X_6, \qquad g_8 = X_4X_5 \cdots X_9. \end{gathered}$$(4)

The Claim

What Is Checked

The claim is stated over the codewords $|s_L\rangle$ of the nine qubit Shor register in (1), and over the logical operators and syndrome observables of (3) and (4), all built by tensor products from the Pauli operators and the identity on single qubits.

About these constructions the library proves that the block cat vector $|000\rangle + (-1)^\sigma|111\rangle$ is nonzero for either bit $\sigma$^✦, so that it normalizes to a pure state, and that a product of pure states and the image of a pure state under a unitary are again unit vectors. It proves that the matrix $\vec n \cdot \vec\sigma$ is Hermitian for real $\vec n$ and the matrix $Z$ unitary, and that tensor products preserve both properties, so that $\bar X$ is a unitary and $\bar Z$ and $g_1, \dots, g_8$ are observables.