Recent years have seen significant advances in quantum/quantum-inspired technologies capable of approximately searching for the ground state of Ising spin Hamiltonians. The promise of leveraging such technologies to accelerate the solution of difficult optimization problems has spurred an increased interest in exploring methods to integrate Ising problems as part of their solution process, with existing approaches ranging from direct transcription to hybrid quantum-classical approaches rooted in existing optimization algorithms. While it is widely acknowledged that quantum computers should augment classical computers, rather than replace them entirely, comparatively little attention has been directed toward deriving analytical characterizations of their interactions. In this paper, we present a formal analysis of hybrid algorithms in the context of solving mixed-binary quadratic programs (MBQP) via Ising solvers. By leveraging an existing completely positive reformulation of MBQPs, as well as a new strong-duality result, we show the exactness of the dual problem over the cone of copositive matrices, thus allowing the resulting reformulation to inherit the straightforward analysis of convex optimization. We propose to solve this reformulation with a hybrid quantum-classical cutting-plane algorithm. Using existing complexity results for convex cutting-plane algorithms, we deduce that the classical portion of this hybrid framework is guaranteed to be polynomial time. This suggests that when applied to NP-hard problems, the complexity of the solution is shifted onto the subroutine handled by the Ising solver.

翻译：摘要：近年来，能够在近似搜索伊辛自旋哈密顿量基态方面取得显著进展的量子/量子启发技术不断涌现。利用此类技术加速复杂优化问题求解的愿景，激发了探索将伊辛问题纳入求解过程的集成方法的研究热潮——现有方法涵盖直接转录及根植于经典优化算法的混合量子-经典方法。尽管量子计算机应增强而非完全替代经典计算机已成为广泛共识，但对其交互作用的解析特征刻画却鲜有关注。本文针对通过伊辛求解器求解混合二元二次规划（MBQP）的混合算法提出形式化分析。通过利用MBQP的完全正重构方法及新的强对偶性结果，我们证明了对偶问题在共正矩阵锥上的精确性，从而使重构形式继承凸优化的简洁分析特性。我们提出采用混合量子-经典割平面算法求解该重构形式。基于现有凸割平面算法的复杂度结论，我们推导出该混合框架中经典部分的多项式时间复杂度保证。这表明当应用于NP困难问题时，问题求解的复杂度被转移至伊辛求解器处理的子程序。