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. We show the exactness of a convex copositive reformulation of MBQPs, 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：近年来，能够近似搜索伊辛自旋哈密顿量基态的量子/量子启发技术取得了显著进展。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整合到求解过程中的方法研究热潮，现有方法从直接转录到基于经典优化算法的混合量子-经典方法不一而足。尽管公认量子计算机应增强而非完全替代经典计算机，但对其相互作用机制的分析性刻画研究相对匮乏。本文针对通过伊辛求解器求解混合二元二次规划（MBQP）的混合算法展开形式化分析。我们证明了MBQP凸共正重构的精确性，使重构问题能继承凸优化简洁分析特性。提出采用混合量子-经典切割平面算法求解该重构问题。基于凸切割平面算法现有复杂度结果，我们推断该混合框架的经典部分可在多项式时间内完成。这表明针对NP困难问题，求解复杂度被转移至伊辛求解器处理的子程序环节。