Symmetry-protected self-correcting quantum code[1] 

A code which admits a restricted notion of thermal stability against symmetric perturbations, i.e., perturbations that commute with a set of operators forming a group \(G\) called the symmetry group.

Given a symmetry group \(G\) and its unitary representation \(S\) on the \(n\)-site physical Hilbert space (in this case, a lattice), an operator \(O\) is \(G\)-symmetric (a.k.a. respects the \(G\) symmetry) if \([S(g),O]=0\) for all \(g\in G\). A symmetry-protected self-correcting memory is a ground-state encoding of an \(n\)-body \(G\)-symmetric geometrically local Hamiltonian whose logical information is recoverable for arbitrary long times in the \(n\to\infty\) limit after a \(G\)-symmetric interaction with a thermal environment at sufficiently low temperature.

Tensor-product symmetries of the form \(S(g)=u(g)^{\otimes n}\), where \(u\) is a unitary representation of \(G\ni g\) on a site, cannot support symmetry-protected self-correction. One can instead use one-form symmetries, i.e., symmetries generated by operators of the form \begin{align} S_{\mathcal{M}}(g)=\bigotimes_{\text{sites}\in\mathcal{M}}u(g), \tag*{(1)}\end{align} where \(\mathcal{M}\) runs over all closed codimension-one submanifolds of the lattice. Recent work further relaxed the requirement so that symmetries need only be enforced on the system's boundaries [2].