Reduction of interaction order in hard combinatorial optimization via conditionally independent degrees of freedom

Abstract

Combinatorial optimization problems have a broad range of applications and map to physical systems with complex dynamics. Among them, the 3-SAT problem is prominent due to its NP-complete nature. In physics terms, its solution corresponds to finding the ground state of a disordered Ising spin Hamiltonian with third-order, or tensor, interactions. The large growth of the number of third-order interactions with number of variables poses technical difficulties for the physical implementation of minimizers. Therefore, researchers have proposed quadratization techniques which reduce the order of the system, however, at the cost of including additional degrees of freedom. Their inclusion induces a drastic slow down in the minimization, which makes such procedures technically infeasible for large problems. In this work, we take a physics approach by employing the renormalization group to create a pairwise interacting system from the original third-order system while preserving the free energy. Our procedure utilizes additional degrees of freedom that exhibit an independent dynamics provided the original degrees of freedom are fixed. A step-wise trace of the extra variables while running the minimization is therefore theoretically manageable, yielding a state-dependent effective interaction. We use the effective interaction to reconstruct the original third-order energy spectrum, as this yields equal scaling of computations-to-ground-state compared to the original tensor formulation. Here, the original degrees of freedom interact with a subsystem that appears to be in a superposition of an exponentially large number of states. In the zero-temperature limit, the superposition concentrates on one state. Our spectrum-engineering techniques reveal new routes toward the ground state of disordered Ising systems, through Markov chains, and allow for efficient technological implementations.