A Copositive Framework for Analysis of Hybrid Ising-Classical Algorithms

TL;DR

This work develops a rigorous framework for hybrid quantum-classical optimization by reformulating mixed-binary quadratic programs ($MBQP$) as copositive programs and proving strong duality with their COP dual. A cutting-plane algorithm is proposed, where a classical solver handles the outer optimization and an Ising-based copositivity oracle supplies separation cuts, effectively shifting the computational burden to the Ising subroutine while guaranteeing polynomial-time complexity for the classical portion. The method is instantiated via a QUBO-based approximation of copositivity checks, enabling practical hybridization with current Ising hardware and simulators; the authors validate the approach on maximum clique benchmarks and compare against MILP and penalty formulations, observing competitive performance under suitable settings. The results illuminate a path for co-designing hardware primitives and optimization algorithms, where copositive optimization serves as a principled bridge between convex analysis and nonconvex combinatorial problems. Overall, the paper contributes a concrete, analyzable blueprint for leveraging Ising solvers within a provably convergent, hybrid architecture with potential practical speedups in NP-hard settings, particularly when the Ising subroutine is the main bottleneck.

Abstract

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.

Figures (12)

• Figure 1: This figure depicts the entire solution process for solving a MBQP of the form \ref{['eq:MBQP']}.
• Figure 1: Figure \ref{['fig:mc_ex']} depicts a small maximum clique example where there are edges between all vertices except $x_4$ and $x_5$. Figure \ref{['fig:mc_ex_mat']} depicts the adjacency matrix of graph \ref{['fig:mc_ex']}'s complement, which has a single edge between vertices $x_4$ and $x_5$. Figure \ref{['fig:mc_ex_sol']} depicts the solution process for the copositive cutting-plane algorithm.
• Figure 1: This figure plots the optimized $\texttt{TTT}_{0.999}$ when Neal was run with 100 sweeps. Optimization produces an order of magnitude speed-up for graphs with 10 nodes but does not result in significant speed-ups for larger graphs.
• Figure 2: This figure plots the time spent on the copositivity checks versus all other operations in the proposed method. The copositivity checks grow exponentially with the number of vertices, while the other operations grow modestly.
• Figure 2: This figure plots the optimal number of sweeps for each of the problem instances. The optimal number of sweeps increases with the graph size. However, graphs of densities $p = 0.25$ and $p = 0.75$ require a comparable number of sweeps for graphs of the same size, while fewer sweeps are required for graphs with density $p = 0.5$

Theorems & Definitions (11)

• Theorem 1: Strong Duality
• Proof 1: Proof Sketch
• Definition 2: Feasibility Problem
• Definition 3: Separation Oracle
• Definition 4: Copositivity Oracle
• Theorem 1: Homogeneous Strong Duality
• Proof 2
• Theorem 2: Inhomogeneous Lower Bound
• Proof 3
• Corollary 3