Minimal constraints in the parity formulation of optimization problems

As a means to solve optimization problems using quantum computers, the problem is typically recast into Ising spin model whose ground-state solution of problem. An alternative formulation Lechner-Hauke-Zoller model, which has form lattice gauge with nearest neighbor 4-body constraints. Here we introduce method find minimal strength constraints are required conserve correct ground-state. Based on this, derive upper and lower bounds for strengths. We that depending class, exponent ranges from linear $\alpha \propto 1$ quadratic 2$ scaling number logical qubits.