Non-Abelian Self-Correcting Quantum Memory and Transversal Non-Clifford Gate beyond the $n^{1/3}$ Distance Barrier

We construct a family of infinitely many new candidate non-Abelian self-correcting topological quantum memories in $D\geq 5+1$ spacetime dimensions without particle excitations using local commuting non-Pauli stabilizer lattice models and field theories of $\mathbb{Z}_2^3$ higher-form gauge fields with nontrivial topological action. We call such non-Pauli stabilizer models magic stabilizer codes. The family of topological orders have Abelian electric excitations and non-Abelian magnetic excitations that obey Ising-like fusion rules and non-Abelian braiding, including Borromean ring type braiding which is a signature of non-Abelian topological order, generalizing the dihedral group $\mathbb{D}_8$ gauge theory in (2+1)D. The simplest example includes a new non-Abelian self-correcting memory in (5+1)D with Abelian loop excitations and non-Abelian membrane excitations. We prove the self-correction property and the thermal stability, and devise a probabilistic local cellular-automaton decoder. We also construct fault-tolerant non-Clifford CCZ logical gate using constant depth circuit from higher cup products in the 5D non-Abelian code. The use of higher-cup products and non-Pauli stabilizers allows us to get an $O(n^{2/5})$ distance overcoming the $O(n^{1/3})$ distance barrier in conventional topological stabilizer codes, including the 3D color code and the 6D self-correcting color code.