I recently came across this paper where the 1d Transverse Field Ising Model (TFIM) with $n$ spins was simulated on a quantum computer. The estimated resources were $n^2$ for the number of gates and the circuit depth was $n \log(n).$ Since the 1d TFIM is equivalent to the 2d Classical Ising Model (CIM) using transfer matrix methods (see here or here), I was wondering
Would the estimated resources for the 2D CIM be the same as the TFIM?
My intuition says we would need more gates since we have a 2D square lattice of size $n^2$ rather than $n$ spins in the TFIM chain. I'm not certain about the circuit depth however. I would really appreciate anyone's thought process here.
