Matrix-model code[1,2] 

Multimode-mode Fock-state bosonic approximate code derived from a matrix model, i.e., a non-Abelian bosonic gauge theory with a large gauge group. The model's degrees of freedom are matrix-valued bosons \(a\), each consisting of \(N^2\) harmonic oscillator modes and subject to an \(SU(N)\) gauge symmetry.

A simple matrix-model code [2] consists of two spatially separated bosons with codewords \begin{align} |\mathcal{I}\rangle :=\prod_{(m,n)\in \mathcal{I} } \frac{\text{Tr}(a_1^{\dagger m}a_2^{\dagger n})}{N^{\frac{m+n}{2}}}|0\rangle_{12}~, \tag*{(1)}\end{align} where \(\cal I\) is some set of integer two-tuples, and \(n,m\geq 0\).

Gauge symmetry is assumed to be enforced in the above model. In other variants [2], gauge symmetry is enforced energetically, requiring a parameter to scale as \(\log(N)\) for polynomial memory lifetime. This translates to the bath coupling being suppressed as \(1/N\).