Operator-algebra (OA) qubit code

Description

An OAQECC family that encompasses ordinary (i.e., subspace) qubit codes, subsystem qubit codes, and hybrid qubit codes using a unified operator-algebraic framework.

A simple example encompassing elements of all three subfamilies encodes a single logical qubit and a single classical bit into a direct sum of two subsystem qubit codes. A quantum subsystem code \(\mathsf{A}_j\otimes\mathsf{B}_j\), with \(\mathsf{A}_j\) the logical qubit factor, and \(\mathsf{B}_j\) the gauge qubit 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 hybrid code when \(\mathsf{B}_{1,2}\) are both trivial, and reduces to an ordinary (i.e., subspace) qubit code when \(\mathsf{B_1}\) and \(\mathsf{A}_{2}\otimes\mathsf{B}_{2}\) are both trivial.