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We develop a global variable substitution method that reduces $n$-variablemonomials in combinatorial optimization problems to equivalent instances withmonomials in fewer variables. We apply this technique to $3$-SAT and analyzethe optimal quantum circuit depth needed to solve the reduced problem using thequantum approximate optimization algorithm. For benchmark $3$-SAT problems, wefind that the upper bound of the circuit depth is smaller when the problem isformulated as a product and uses the substitution method to decompose gatesthan when the problem is written in the linear formulation, which requires nodecomposition.

Globally optimizing QAOA circuit depth for constrained optimization problems