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.

翻译：QUBO可以被视为优化问题的一种通用语言。QUBO吸引了特别关注,因为它们可以用量子硬件来解决,如运行QAOA的量子喷射器或量子门计算机。在本文中,我们提出了两个新型的QUBO配方,其规模比现有方法大得多,用于美元SAT和汉密尔顿周期。对于美元SAT,我们把QUBO矩阵的增长从(k)美元减少到了O(log(k)美元)。对于汉密尔顿周期,矩阵不再像目前那样在节点数量上以二次方式增长,而是线性地增长。我们把这两种配方不作为数学表达,因为大多数QUBO配方都是数学表达的,而是作为促进设计更复杂的QUBO配方,并允许在更大和更复杂的QUBO配方中方便再利用的元-方。