Hybrid QECC[1–4]

Description

A quantum code which encodes both quantum and classical information.

Hybrid QECCs arise as the \(e=0\) subclass of the EACQ formalism, i.e., the classically enhanced quantum codes that do not require pre-shared entanglement [4].

In general, a different quantum code \(\mathsf{C}_j\) is associated with each classical value \(j \in \{0, 1, \ldots, l-1\}\), and the Hilbert space decomposes as \begin{align} \mathsf{H} = \bigoplus_{j=0}^{l-1} \mathsf{C}_j \oplus \mathsf{C}^{\perp}~, \tag*{(1)}\end{align} where \(\mathsf{C}^{\perp}\) is the combined error space of all \(l\) codes. The simplest example encodes a single qubit and a single classical bit (\(l = 2\)): a different quantum code is associated with each value \(j \in \{0,1\}\), giving \(\mathsf{H} = \mathsf{C}_0 \oplus \mathsf{C}_1 \oplus \mathsf{C}^{\perp}\). The error-correction conditions require the Knill-Laflamme conditions to hold within each quantum code subspace \(\mathsf{C}_j\) [5; Eq. (3)], and that error operators map different code subspaces to mutually orthogonal subspaces [5; Eq. (4)].