Globally optimizing QAOA circuit depth for constrained optimization problems

TL;DR

This paper introduces a global variable substitution method that reduces the complexity of combinatorial optimization problems, enabling more efficient quantum approximate optimization algorithm (QAOA) circuit design for problems like 3-SAT.

Contribution

The paper presents a novel variable substitution technique that decreases monomial variables and optimizes QAOA circuit depth for constrained problems like 3-SAT.

Findings

Reduced circuit depth when using the substitution method for 3-SAT.

Lower upper bounds on quantum circuit depth with problem decomposition.

Decomposition improves efficiency over linear formulations.

Abstract

We develop a global variable substitution method that reduces $n$-variable monomials in combinatorial optimization problems to equivalent instances with monomials in fewer variables. We apply this technique to $3$-SAT and analyze the optimal quantum circuit depth needed to solve the reduced problem using the quantum approximate optimization algorithm. For benchmark $3$-SAT problems, we find that the upper bound of the circuit depth is smaller when the problem is formulated as a product and uses the substitution method to decompose gates than when the problem is written in the linear formulation, which requires no decomposition.