  While LDPC codes have been demonstrated with desirable error correcting
properties, this has come at a cost of diverging from the geometrical
constraints of many hardware platforms. Viewing codes as the groundspace of a
Hamiltonian, we consider engineering a simulation Hamiltonian reproducing some
relevant features of the code. Techniques from Hamiltonian simulation theory
are used to build a simulation of LDPC codes using only 2D nearest-neighbour
interactions at the cost of an energy penalty polynomial in the system size. We
derive guarantees for the simulation that allows us to approximately reproduce
the ground state of the code Hamiltonian, approximating a $[[N,
\Omega(\sqrt{N}), \Omega(\sqrt{N})]]$ code in 2D. The key ingredient is a new
constructive tool to simulate an $l$-long interaction between two qubits by a
1D chain of $l$ nearest-neighbour interacting qubits using $\mathrm{poly}( l)$
interaction strengths. This is an exponential advantage over the existing
gadgets for this routine which facilitates the first $\epsilon$-simulation of
\emph{arbitrary sparse} Hamiltonian on $n$ qubits with a Hamiltonian on a 2D
lattice of $O(n^2)$ qubits with interaction strengths scaling as
$O\left(\mathrm{poly}(n,1/\epsilon)\right)$.
