The Exercise

Shor’s code stores one qubit in nine, grouped into three blocks of three. Up to normalization, each block is in one of the two states $|000\rangle + |111\rangle$ and $|000\rangle - |111\rangle$, and a phase flip, the Pauli operator $Z$ acting on any one qubit of a block, takes one to the other. A phase flip is found by comparing the signs of the blocks, and two comparisons suffice: the operator $X_1 X_2 X_3 X_4 X_5 X_6$, with $X$ acting on each of the first six qubits, compares blocks one and two^✦, and $X_4 X_5 X_6 X_7 X_8 X_9$ compares blocks two and three.^✦

The exercise is to show that these two observables measure the phase flip syndrome, the pair of values that tells which block, if any, has suffered a phase flip.

Nine Qubits in Three Blocks

A quantum system $S$ is a physical system described by its state space, a finite dimensional complex Hilbert space.^✦ 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.^✦ The qubit $Q$ is the two level system with state space $\mathbb{C}^2$.^✦ A block is the three qubit register $R = Q \otimes Q \otimes Q$^✦, and Shor’s nine qubit register is $R \otimes R \otimes R$.^✦

A pure state $|\psi\rangle$ of $S$ is a vector of unit norm in the state space of $S$.^✦ For pure states $|\psi\rangle$ of $S$ and $|\varphi\rangle$ of $T$, the product $|\psi\rangle \otimes |\varphi\rangle$ is the pure state of $S \otimes T$ whose vector is the tensor product of the two vectors.^✦ The basis states $|0\rangle$ and $|1\rangle$ of the qubit are the standard basis vectors of $\mathbb{C}^2$^✦, and for three bits $q_1, q_2, q_3$ the basis state of a block is $|q_1 q_2 q_3\rangle = |q_1\rangle \otimes |q_2\rangle \otimes |q_3\rangle$.^✦ Given a nonzero vector $|v\rangle$ in the state space of $S$, its normalized state is the pure state $|v\rangle / \sqrt{\langle v | v \rangle}$.^✦

For a bit $\sigma$, the block cat vector is $|000\rangle + (-1)^\sigma |111\rangle$ in the state space of $R$^✦, and the block cat state $|c_\sigma\rangle$ is its normalized state, $$|c_\sigma\rangle = \frac{1}{\sqrt{2}} \bigl( |000\rangle + (-1)^\sigma |111\rangle \bigr) .$$(1)^✦ Three bits $\sigma_1, \sigma_2, \sigma_3$ give the product of three cat blocks, the pure state of the nine qubit register $$|\Psi_{\sigma_1 \sigma_2 \sigma_3}\rangle = |c_{\sigma_1}\rangle \otimes |c_{\sigma_2}\rangle \otimes |c_{\sigma_3}\rangle ,$$(2)^✦ in which block $k$ carries the sign $(-1)^{\sigma_k}$.

The Syndrome Observables

An observable $A$ on $S$ is a self adjoint operator on the state space of $S$, $A^\dagger = A$.^✦ The identity observable $I$ on $S$ is the identity operator on its state space.^✦ For observables $A$ on $S$ and $B$ on $T$, the product observable $A \otimes B$ on $S \otimes T$ has as its operator the tensor product of the two operators; it measures $A$ on the first factor and $B$ on the second.^✦

The Pauli matrices are $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},$$(3) and for a real vector $\vec{n}$ in three dimensions the matrix $\vec{n} \cdot \vec{\sigma} = n_x X + n_y Y + n_z Z$ is the operator of the Pauli observable $\vec{n} \cdot \vec{\sigma}$ on the qubit. The observable $X$ is the Pauli observable for the unit vector $\hat{n} = (1, 0, 0)$, the direction of the first axis.^✦

The block observable $X \otimes X \otimes X$ on $R$ applies $X$ to each of the three qubits.^✦ The two syndrome observables on the nine qubit register are the product observables built from the block observable $X \otimes X \otimes X$ on $R$ and the identity observable $I$ on $R$: $$X_1 X_2 X_3 X_4 X_5 X_6 = (X \otimes X \otimes X) \otimes (X \otimes X \otimes X) \otimes I ,$$(4)^✦ $$X_4 X_5 X_6 X_7 X_8 X_9 = I \otimes (X \otimes X \otimes X) \otimes (X \otimes X \otimes X) .$$(5)^✦

A pure state $|\psi\rangle$ of $S$ is an eigenstate of an observable $A$ on $S$ with eigenvalue $\mu$, a real number, when $A |\psi\rangle = \mu |\psi\rangle$.^✦

The Claim

What Is Checked

The claim is stated over the product states (2) of three block cat states on the nine qubit register, and over the syndrome observables (4) and (5).

The library proves that the block cat vector $|000\rangle + (-1)^\sigma |111\rangle$ is nonzero^✦, so that its normalized state is defined, and that a product of pure states has unit norm, the norm of a product vector being the product of the norms. On the side of the observables, the matrix $\vec{n} \cdot \vec{\sigma}$ is Hermitian and a tensor product of self adjoint operators is self adjoint, so $X \otimes X \otimes X$ and both syndrome operators are observables.