Algorithmic QUBO Formulations for k-SAT and Hamiltonian Cycles

TL;DR

This paper introduces two scalable QUBO formulations for k-SAT and Hamiltonian Cycles, enabling more efficient problem encoding for quantum optimization hardware, and presents them as meta-algorithms for broader application.

Contribution

The paper presents novel, scalable QUBO formulations for k-SAT and Hamiltonian Cycles, improving upon existing methods and providing meta-algorithms for designing complex QUBO problems.

Findings

QUBO matrix growth for k-SAT reduced from O(k) to O(log(k))

QUBO matrix for Hamiltonian Cycles grows linearly in edges and logarithmically in nodes

Formulations serve as meta-algorithms for complex QUBO design

Abstract

Quadratic unconstrained binary optimization (QUBO) can be seen as a generic language for optimization problems. QUBOs attract particular attention since they can be solved with quantum hardware, like quantum annealers or quantum gate computers running QAOA. In this paper, we present two novel QUBO formulations for $k$-SAT and Hamiltonian Cycles that scale significantly better than existing approaches. For $k$-SAT we reduce the growth of the QUBO matrix from $O(k)$ to $O(log(k))$. For Hamiltonian Cycles the matrix no longer grows quadratically in the number of nodes, as currently, but linearly in the number of edges and logarithmically in the number of nodes. We present these two formulations not as mathematical expressions, as most QUBO formulations are, but as meta-algorithms that facilitate the design of more complex QUBO formulations and allow easy reuse in larger and more complex QUBO formulations.