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

A Copositive Framework for Analysis of Hybrid Ising-Classical Algorithms