Self-Correcting 3D Surface Code with Growing Local Dimension

We construct self-correcting 3D surface codes over $\mathbb Z_q$ with local dimension $q$ growing with system size. The model preserves the local geometry and stabilizer support of the ordinary 3D surface code and remains translation invariant in the bulk, while modifying only the stabilizer Pauli powers. We use the Lee metric to characterize error weight and consider local Metropolis dynamics with single-qudit $X^{\pm1}$ and $Z^{\pm1}$ updates, each changing the energy by at most a constant independent of $q$ and system size $N$. We prove that, below the critical temperature, when the local dimension scales as $q=Θ(N)$, the memory times in both error sectors grow at least as $\exp\left[Ω(N^{1/3})\right]$. Our numerical results further support these analytical results. Together, these results establish growing local Hilbert-space dimension as a new resource for designing self-correcting quantum memories while retaining simple local geometry and local thermal dynamics.