The Exercise

The three qubit bit flip code stores one qubit in three, in the subspace spanned by $\lvert 000\rangle$ and $\lvert 111\rangle$. The noise flips each of the three qubits independently with probability $p$, and it is described by four operation elements: one for the event that no qubit flips, and one for each event in which exactly one qubit flips. Exercise 10.7 asks one to verify that this code and this noise satisfy the quantum error correction conditions.

Three Qubits

A quantum system $S$ is a physical system whose state space is 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 the state spaces of $S$ and $T$.^✦ The qubit $Q$ is the system whose state space is $\mathbb{C}^2$^✦, and the three qubit register $R = Q \otimes (Q \otimes Q)$ is the composite of three qubits, the system that carries the code.^✦

A pure state $\lvert\psi\rangle$ of a system $S$ is a vector of unit norm in the state space of $S$.^✦ For pure states $\lvert\psi\rangle$ of $S$ and $\lvert\varphi\rangle$ of $T$, the product $\lvert\psi\rangle \otimes \lvert\varphi\rangle$ is the pure state of $S \otimes T$ whose vector is the tensor product of the two vectors.^✦ The basis states $\lvert 0\rangle$ and $\lvert 1\rangle$ of the qubit are the pure states whose vectors are the standard basis vectors $e_0$ and $e_1$ of $\mathbb{C}^2$.^✦ For bits $q_1, q_2, q_3$ the basis state of the register is their product, $$\lvert q_1 q_2 q_3\rangle = \lvert q_1\rangle \otimes \bigl(\lvert q_2\rangle \otimes \lvert q_3\rangle\bigr),$$(1)^✦ and the eight of them form the computational basis of $R$.

Bit Flips

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

The Pauli matrices are $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},$$(2)^✦ and, writing $\vec{\sigma} = (X, Y, Z)$ for the Pauli vector, for a real vector $\vec{n} = (n_x, n_y, n_z)$ in three dimensions the Pauli observable $\vec{n} \cdot \vec{\sigma} = n_x X + n_y Y + n_z Z$ is the observable of the qubit with that matrix as its operator. The bit flip $X$ is the Pauli observable along $\hat{n} = (1, 0, 0)$.^✦ On the register, the flip of qubit $j$ acts as $X$ on that qubit and as the identity on the other two: $$X_1 = X \otimes (I \otimes I), \qquad X_2 = I \otimes (X \otimes I), \qquad X_3 = I \otimes (I \otimes X).$$(3)^✦

The Code and the Noise

The three qubit bit flip code has four syndromes, one for no flip and one for a flip of each qubit, labelled by $i \in \{0, 1, 2, 3\}$, and each carries a pair $(a_i, b_i)$ of three bit strings: $(000, 111)$ for $0$, the two codewords, and $(100, 011)$, $(010, 101)$ and $(001, 110)$ for $1$, $2$ and $3$, the codewords with bit $i$ flipped.^✦ For a label $i$, the syndrome projector $P_i$ is the operator on the state space of $R$ that is the sum of the projectors onto the basis states (1) of its pair, $$P_i = \lvert a_i\rangle\langle a_i\rvert + \lvert b_i\rangle\langle b_i\rvert.$$(4)^✦ The code projector $P_0$ of the three qubit bit flip code is the projector onto its code space.

The noise is built from the error operators on the state space of $R$, $$T_0 = I, \qquad T_j = X_j \quad (j = 1, 2, 3),$$(5)^✦ and from the coefficients $$c_0 = \sqrt{(1-p)^3}, \qquad c_1 = c_2 = c_3 = \sqrt{p(1-p)^2}.$$(6)^✦ For $0 \le p \le 1$, $c_0^2$ is the probability that no qubit flips and $c_j^2$ the probability that qubit $j$ flips while the other two do not. The noise elements $E_i$ of the three qubit bit flip noise are the operators on the state space of $R$ obtained by weighting the error operators with these coefficients, $$E_i = c_i\, T_i, \qquad i = 0, 1, 2, 3.$$(7)^✦

The Claim

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

What Is Checked

The claim is stated over the code projector $P_0$ of (4) and the noise elements of (7). About these constructions the library proves that each flip $X_j$ of (3) is an observable, self adjoint as a tensor product of self adjoint operators, the bit flip $X$ being self adjoint because its matrix is Hermitian; and that each basis state $\lvert q_1 q_2 q_3\rangle$ of (1) has unit norm, the norm of a product vector being the product of the norms.