The Exercise

Shor’s code stores one qubit in nine, grouped into three blocks of three: qubits $1$ to $3$, $4$ to $6$ and $7$ to $9$. Each of its two codewords $|0_L\rangle$ and $|1_L\rangle$ puts every block in a cat state, $|000\rangle + |111\rangle$ for $|0_L\rangle$ and $|000\rangle - |111\rangle$ for $|1_L\rangle$. Eight operators on the nine qubits go with the code: six products $Z_i Z_j$ of Pauli $Z$ on two neighbouring qubits of one block, and two products of Pauli $X$ on all six qubits of two neighbouring blocks, $X_1 \cdots X_6$ and $X_4 \cdots X_9$. The exercise is to show that each of the eight leaves both codewords unchanged.

A Quantum System

A quantum system $S$ is a physical system described by its state space, a finite dimensional complex Hilbert space.^✦ Every finite dimensional complex Hilbert space $H$ is the state space of a quantum system; the qubit $Q$, the two level system, is the one whose state space is $\mathbb{C}^2$. The composite system $S \otimes T$ of two 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 vector $|\psi\rangle$ of unit norm in the state space of $S$, determined physically up to a global phase.^✦ A nonzero vector $|v\rangle$ in that space gives the normalized pure state $|v\rangle / \lVert v \rVert$.^✦ 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 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 $\mathbb{C}^2$.^✦ The three qubit register $R = Q \otimes Q \otimes Q$ is the composite of three qubits.^✦ Its basis state $|q_1 q_2 q_3\rangle$, for bits $q_1$, $q_2$ and $q_3$, is the product $$|q_1 q_2 q_3\rangle = |q_1\rangle \otimes |q_2\rangle \otimes |q_3\rangle .$$(1) The nine qubit Shor register is the composite $R \otimes R \otimes R$ of three copies of $R$, one for each block of the code.^✦

Observables and Eigenstates

An observable $A$ on $S$ is a self adjoint operator on the state space of $S$, $A^\dagger = A$; its real spectrum is the set of possible measured values.^✦ A pure state $\psi$ of $S$ is an eigenstate of $A$ with eigenvalue $\mu$, a real number, when $$A|\psi\rangle = \mu|\psi\rangle .$$(2)^✦ The identity observable $I$ on $S$ is the observable whose operator is the identity.^✦ For an observable $A$ on $S$ and an observable $B$ on $T$, the tensor product $A \otimes B$ is the observable on $S \otimes T$ whose operator is the tensor product of the two operators; it measures $A$ on the first factor and $B$ on the second.^✦

On the qubit, observables are built from the Pauli matrices, complex $2 \times 2$ matrices with rows and columns indexed by $0$ and $1$: $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$(3) The Pauli vector is $\vec{\sigma} = (X, Y, Z)$. For a real vector $\vec{n} = (n_x, n_y, n_z)$ the matrix $\vec{n} \cdot \vec{\sigma}$ is $n_x X + n_y Y + n_z Z$^✦, and the Pauli observable $\vec{n} \cdot \vec{\sigma}$ is the observable on the qubit with this matrix as its operator on $\mathbb{C}^2$.^✦ The observables $X$ and $Z$ are the Pauli observables for $\hat{n} = (1, 0, 0)$ and $\hat{n} = (0, 0, 1)$.

On the three qubit register, the observables $Z_1 Z_2$ and $Z_2 Z_3$, which compare neighbouring qubits and detect a bit flip, and the observable $X_1 X_2 X_3$ that flips all three qubits are the tensor products $$Z_1 Z_2 = Z \otimes Z \otimes I, \qquad Z_2 Z_3 = I \otimes Z \otimes Z, \qquad X \otimes X \otimes X,$$(4) with $I$ the identity observable on one qubit. $Z_1 Z_2$ compares the first qubit with the second, and $Z_2 Z_3$ the second with the third.

The Shor Codewords

For a bit $\sigma$, the block cat vector $|000\rangle + (-1)^\sigma |111\rangle$ is a vector in the state space of the three qubit register, built from the basis states (1) whose bits are all $0$ and all $1$.^✦ The block cat state $|c_\sigma\rangle$ of the three qubit register $R$ is the block cat vector divided by its norm: $$|c_\sigma\rangle = \frac{1}{\sqrt{2}} \bigl( |000\rangle + (-1)^\sigma |111\rangle \bigr).$$(5) For bits $\sigma_1$, $\sigma_2$ and $\sigma_3$, the product of three cat states is the pure state $|c_{\sigma_1}\rangle \otimes |c_{\sigma_2}\rangle \otimes |c_{\sigma_3}\rangle$ of the Shor register, one cat state per block.^✦ The Shor codeword $|s_L\rangle$, for a bit $s$, is the product with the same sign in all three blocks: $$|s_L\rangle = |c_s\rangle \otimes |c_s\rangle \otimes |c_s\rangle = \frac{1}{2\sqrt{2}} \bigl( |000\rangle + (-1)^s |111\rangle \bigr)^{\otimes 3}.$$(6)^✦

The Eight Generators

The eight generators $g_1, \dots, g_8$ of the code are observables on the Shor register, each the tensor product of one observable per block, an $I$ outside the brackets being the identity observable on a whole block: $$\begin{aligned} g_1 &= Z_1 Z_2 = (Z \otimes Z \otimes I) \otimes I \otimes I, \\ g_2 &= Z_2 Z_3 = (I \otimes Z \otimes Z) \otimes I \otimes I, \\ g_3 &= Z_4 Z_5 = I \otimes (Z \otimes Z \otimes I) \otimes I, \\ g_4 &= Z_5 Z_6 = I \otimes (I \otimes Z \otimes Z) \otimes I, \\ g_5 &= Z_7 Z_8 = I \otimes I \otimes (Z \otimes Z \otimes I), \\ g_6 &= Z_8 Z_9 = I \otimes I \otimes (I \otimes Z \otimes Z), \\ g_7 &= X_1 \cdots X_6 = (X \otimes X \otimes X) \otimes (X \otimes X \otimes X) \otimes I, \\ g_8 &= X_4 \cdots X_9 = I \otimes (X \otimes X \otimes X) \otimes (X \otimes X \otimes X). \end{aligned}$$(7)^✦

The Claim

What Is Checked

The claim is stated over the two Shor codewords (6), products of three normalized block cat states, and the eight generators (7), tensor products of the observables $X$, $Z$ and $I$, with eigenstate taken in the sense of (2).

About these constructions the library proves that the block cat vector $|000\rangle + (-1)^\sigma |111\rangle$ is nonzero for both values of $\sigma$^✦, so that it can be normalized, and that the product of two pure states is again of unit norm, the norm of a product vector being the product of the norms. It proves as well that the matrix $\vec{n} \cdot \vec{\sigma}$ is Hermitian, so that every Pauli observable is self adjoint, and that a tensor product of observables is self adjoint, as a tensor product of Hermitian operators. The identity observable $I$ is self adjoint as well, the identity being its own adjoint.