Operator-algebra QECC (OAQECC)[1–6] 

A code family that encompasses ordinary (i.e., subspace) codes, subsystem codes, classical-quantum codes, and hybrid codes using a unified operator-algebraic framework.

A simple example encompassing elements of all subfamilies encodes quantum information and a single classical bit into a direct sum of two subsystem codes. A quantum subsystem code \(\mathsf{A}_j\otimes\mathsf{B}_j\), with \(\mathsf{A}_j\) the logical factor associated with the quantum information, and \(\mathsf{B}_j\) the gauge factor, is associated with each of the two values \(j\in\{0,1\}\) of the classical bit. The corresponding decomposition of the Hilbert space \(\mathsf{H}\) is \begin{align} \mathsf{H}=(\mathsf{A}_{1}\otimes\mathsf{B}_{1})\oplus(\mathsf{A}_{2}\otimes\mathsf{B}_{2})\oplus\mathsf{C}^{\perp}~, \tag*{(1)}\end{align} where \(\mathsf{C}^\perp\) is the combined error space of both codes. The above code reduces to a subsystem code when \(\mathsf{A}_{2}\otimes\mathsf{B}_{2}\) is trivial, reduces to a classical-quantum code when \(\mathsf{A}_{1,2}\) are both trivial, reduces to a hybrid code when \(\mathsf{B}_{1,2}\) are both trivial, and reduces to an ordinary (i.e., subspace) code when \(\mathsf{B_1}\) and \(\mathsf{A}_{2}\otimes\mathsf{B}_{2}\) are both trivial.

In general, an OAQECC code is determined by a finite dimensional \(C^*\) algebra \(\mathcal{A}\) of operators on \(\mathsf{H}\). This logical algebra induces a decomposition of the Hilbert space as \begin{align}\mathsf{H} = \bigoplus_\gamma \mathsf{A}_\gamma \otimes \mathsf{B}_\gamma,\tag*{(2)}\end{align} with respect to which \(\mathcal{A}\) takes the form \begin{align}\mathcal{A} = \bigoplus_\gamma I_\gamma \otimes \mathcal{L}(\mathsf{B}_\gamma),\tag*{(3)}\end{align} where \(\mathcal{L}(\mathsf{B}_\gamma)\) denotes the full set of linear maps on \(\mathsf{B}_\gamma\). The \(\mathsf{A}_j\) factors can be used to store quantum information, \(\gamma\) indexes the block structure of the code, while \(\mathsf{B}_j\) determine its gauge structure. Together, the above forms the most general form of an information preserving structure [7,8].