The Exercise

Let $C_1$ and $C_2$ be classical linear codes of length $n$, subspaces of $\mathbb{Z}_2^n$, and write $|w\rangle$ for the computational basis state of $n$ qubits labelled by the bit string $w$. For a string $x \in \mathbb{Z}_2^n$, the coset state $|x + C_2\rangle$ is the pure state of $n$ qubits spread uniformly over the coset $x + C_2$, $$|x + C_2\rangle = \frac{1}{\sqrt{|C_2|}} \sum_{y \in C_2} |x + y\rangle .$$(1)^✦ For $x \in C_1$ these are the codewords of the Calderbank, Shor and Steane (CSS) code $\mathrm{CSS}(C_1, C_2)$.^✦

Fix two bit strings $u$ and $v$. Shifting every term by $v$ and giving the term of $y$ the sign $(-1)^{u \cdot y}$, where $u \cdot y = \sum_i u_i y_i \bmod 2$, turns the coset state into the twisted coset state $$|x + C_2\rangle_{u,v} = \frac{1}{\sqrt{|C_2|}} \sum_{y \in C_2} (-1)^{u \cdot y}\, |x + y + v\rangle ,$$(2)^✦ and for $x \in C_1$ these are the codewords of a code $\mathrm{CSS}_{u,v}(C_1, C_2)$.^✦ The exercise is to show that this code has the same error correcting properties as $\mathrm{CSS}(C_1, C_2)$.

The Register

The register of the code is the group system $\mathbb{C}[\mathbb{Z}_2^n]$ of $n$ qubits: the quantum system whose state space is the space of complex functions on the bit strings $w \in \mathbb{Z}_2^n$, a finite dimensional complex Hilbert space with the orthonormal computational basis $\{|w\rangle\}$, one vector for each string.^✦

A state $\rho$ of the register is a positive operator of unit trace on its state space.^✦ A pure state $|\psi\rangle$ of the register is a unit vector in its state space^✦, and it determines the state $|\psi\rangle\langle\psi|$, the projector onto that vector.^✦

A unitary $U$ of the register is an operator on its state space with $U^\dagger U = U U^\dagger = 1$, where $U^\dagger$ is the adjoint of $U$.^✦ It acts on a state $\rho$ by conjugation, $$\rho \mapsto U \rho U^\dagger ,$$(3)^✦ and two unitaries $U$ and $V$ compose to the product $UV$, which applies $V$ first.^✦

Pauli Strings

For a bit string $v$, the bit flip string $X^v$ is the unitary of the register that adds $v$ to each basis string, $$X^v |w\rangle = |w + v\rangle .$$(4)^✦ The phase flip string $Z^u$ of a bit string $u$ is the unitary diagonal in the computational basis with $$Z^u |w\rangle = (-1)^{u \cdot w}\, |w\rangle .$$(5)^✦ The Pauli string of $u$ and $v$ is the unitary $$W = Z^u X^v .$$(6)

Codes and Correction

The CSS code $\mathrm{CSS}(C_1, C_2)$ is the set of states of the register given by the coset states (1) of the strings of $C_1$, $$\mathrm{CSS}(C_1, C_2) = \bigl\{\, |x + C_2\rangle\langle x + C_2| \;:\; x \in C_1 \,\bigr\} ,$$(7)^✦ and the twisted CSS code $\mathrm{CSS}_{u,v}(C_1, C_2)$ is the set given in the same way by the twisted coset states (2), $$\mathrm{CSS}_{u,v}(C_1, C_2) = \bigl\{\, |x + C_2\rangle_{u,v}\,{}_{u,v}\langle x + C_2| \;:\; x \in C_1 \,\bigr\} .$$(8)^✦

Let $\mathcal{E}$ and $\mathcal{R}$ be maps taking states of the register to states, and let $C$ be a set of its states. The recovery $\mathcal{R}$ corrects the noise $\mathcal{E}$ on $C$ when $\mathcal{E}$ followed by $\mathcal{R}$ leaves every state of $C$ unchanged, $$\mathcal{R}(\mathcal{E}(\rho)) = \rho \quad \text{for every } \rho \in C .$$(9)^✦

The Claim

What Is Checked

The claim is stated over the two codes as sets of states of the register, (7) and (8), for any linear codes $C_1$ and $C_2$ and any strings $u$ and $v$; over the Pauli string $W$ of (6) and its adjoint, acting on states by conjugation (3); and over any maps $\mathcal{E}$ and $\mathcal{R}$ on the states of the register, with correction in the sense of (9).

The library shows that the factor $1/\sqrt{|C_2|}$ makes each coset state, plain or twisted, a unit vector: the strings $x + y$, and likewise $x + y + v$, are distinct for distinct $y \in C_2$, and the signs $(-1)^{u \cdot y}$ have modulus one. The phase flip string $Z^u$ is a unitary because its diagonal entries $\pm 1$ have modulus one. The bit flip string $X^v$ permutes an orthonormal basis, so it is an isometry of the state space onto itself, and such an isometry is a unitary^✦; $W$ is a unitary as a product of unitaries. The adjoint of a unitary is a unitary, and conjugating a state by a unitary gives a state.