Information storage capacity of discrete spin systems

What is the limit of information storage capacity of discrete spin systems? To answer this question, we study classical error-correcting codes which can be physically realized as the energy ground space of gapped local Hamiltonians. For discrete spin systems on a D-dimensional lattice governed by local frustration-free Hamiltonians, the following bound is known to hold; kd 1/D ≤ O(n) where k is the number of encodable logical bits, d is the code distance, and n is the total number of spins in the system. Yet, previously found codes were far below this bound and it remained open whether there exists an error-correcting code which saturates the bound or not. Here, we give a construction of local spin systems which saturate the bound asymptotically with k ∼ O(L D−1) and d ∼ O(L D−) for an arbitrary small > 0 where L is the linear length of the system. Our model borrows an idea from a fractal geometry arising in Sierpinski triangle.