Group GKP code[1]

Description

Group-based quantum code whose construction is based on nested subgroups \(H\subset K \subset G\). The group GKP code was originally formulated as an extension of the GKP code construction to other group-valued spaces. In other words, the only requirement to construct group GKP codes is that the configuration space \(G\) is a group under some operation.

Group GKP codes are stabilized by \(X\)-type group-based right-multiplication error operators representing \(H\), all \(Z\)-type operators that are constant on \(K\), and any left-multiplication error operators that are constant on the codespace. The \(Z\)-type operators are non-unitary matrix product operators for non-Abelian groups. The codes’ logical subspace is spanned by basis states that are equal superpositions of elements of cosets of \(H\) in \(K\), and can be finite- or infinite-dimensional depending on the chosen groups.

The construction encompasses all Pauli-type CSS codes since those are Abelian group GKP codes; relevant \(G\) and \(H\) are tabulated in Tab. I. An \(n\) Galois-qubit CSS code corresponds to the \(GF(q)^{k_1} \subseteq GF(q)^{k_2} \subset GF(q)^{n}\) group construction, where \(k=k_2-k_1\), and where the group operation is addition; this construction should be extendable to additive \(q\)-ary codes since those are also groups under addition. An \([[n,k,d]]_{\mathbb{R}}\) analog CSS code corresponds to the \(\mathbb{R}^{ k_1} \subseteq \mathbb{R}^{ k_2} \subset \mathbb{R}^{n}\) group construction, where \(k=k_2-k_1\). A single-mode qubit GKP CSS code corresponds to the \(2\mathbb{Z}\subset\mathbb{Z}\subset\mathbb{R}\) group construction, and multimode GKP CSS codes can be similarly described. Oscillator-into-oscillator GKP CSS codes for \(n\) modes correspond to subgroups \(\mathbb{Z}^m\) for \(m<n\). Rotor GKP codes correspond to the \(\mathbb{Z}_{k_1} \subseteq \mathbb{Z}_{k_2} \subset U(1)\) group construction, where \(k=k_2/k_1\).