Recent hardware advances in quantum and quantum-inspired annealers promise substantial speedup for solving NP-hard combinatorial optimization problems compared to general-purpose computers. These special-purpose hardware are built for solving hard instances of Quadratic Unconstrained Binary Optimization (QUBO) problems. In terms of number of variables and precision of these hardware are usually resource-constrained and they work either in Ising space {-1,1} or in Boolean space {0,1}. Many naturally occurring problem instances are higher-order in nature. The known method to reduce the degree of a higher-order optimization problem uses Rosenberg's polynomial. The method works in Boolean space by reducing the degree of one term by introducing one extra variable. In this work, we prove that in Ising space the degree reduction of one term requires the introduction of two variables. Our proposed method of degree reduction works directly in Ising space, as opposed to converting an Ising polynomial to Boolean space and applying previously known Rosenberg's polynomial. For sparse higher-order Ising problems, this results in a more compact representation of the resultant QUBO problem, which is crucial for utilizing resource-constrained QUBO solvers.

中文翻译：

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高阶二元优化问题的压缩二次化

与通用计算机相比，量子和量子启发式退火器的最新硬件进步有望大大加快解决 NP 难组合优化问题的速度。这些专用硬件专为解决二次无约束二元优化 (QUBO) 问题的困难实例而构建。就变量数量和精度而言，这些硬件通常是资源受限的，它们在伊辛空间 {-1,1} 或布尔空间 {0,1} 中工作。许多自然发生的问题实例本质上是高阶的。降低高阶优化问题的程度的已知方法使用罗森伯格多项式。该方法通过引入一个额外变量来减少一项的次数，从而在布尔空间中起作用。在这项工作中，我们证明，在 Ising 空间中，一项的度数减少需要引入两个变量。我们提出的度数减少方法直接在 Ising 空间中起作用，而不是将 Ising 多项式转换为布尔空间并应用先前已知的 Rosenberg 多项式。对于稀疏的高阶 Ising 问题，这会导致结果 QUBO 问题的更紧凑表示，这对于利用资源受限的 QUBO 求解器至关重要。