Let \(V=\mathbb{C}^{2}\). Let \(G \subset U(V)\) be a finite group of unitary operators defined as \[\begin{align} \left\langle \begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & -1\end{bmatrix} \begin{bmatrix} 0 & -i \\ i & 0\end{bmatrix} \right\rangle. \end{align}\] These three matrices are called Pauli matrices and they have an interpretations as a bit error, phase error and a combination of bit and phase error respectively.
Note that \(G^{n}\) acts on \(V^{\otimes n}\) naturally. We can identify \(G^{n} \subseteq U(V^{\otimes n}) = U(2^{n})\). \(G^{n}\) is a finite group, and the weight function \[\begin{align} w : G^n & \rightarrow \mathbb{Z}_{\ge 0} \\ (g_1,g_2,\ldots,g_n) & \mapsto \#\left\{ i \in \{ 1,2,\ldots,n\} \ | \ g_i \neq \begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix}\right\} . \end{align}\]
We can define the Pauli group \(\mathcal{G}_{n} = \{ \pm 1, \pm i\} G^{n}\). That extra scalar in \(\{ \pm 1, \pm i\}\) is called the global phase. We can extend the weight function \(w\) on \(\mathcal{G}_{n}\) by ignoring the effect on the global phase.
An \([n,k,d]\) quantum error correcting code is a subspace \(W \subset V^{\otimes n}\) such that \(\dim_{\mathbb{C}} W = 2^{k}\) and for every \(g \in \mathcal{G}_{n}\) such that \(w(g) \le d\), we have \[\begin{align} v \in W \Rightarrow gv - v\in W^{\perp}. \end{align}\]
There is also the book [1] about this subject. Things written by Daniel Gottesman are very accessible for mathematicians, in particular the PhD thesis [2].
In classical binary error-correcting codes, each information is a bit. The errors are propagated as binary random variables adding randomly at coordinates.
Here, the information is a qubit, which is seen as one copy of \(V\). The errors act as Pauli matrices on \(V\).
Check out https://earltcampbell.com/2016/09/26/the-smallest-interesting-colour-code/
This page was updated on December 1, 2022.
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