Constraint-Aware Quantum Optimization via Hamming Weight Operators

TL;DR

This paper introduces Hamming Weight Operators for quantum optimization, enabling constraint satisfaction within feasible subspaces, leading to more efficient and scalable quantum algorithms for practical constrained problems.

Contribution

The paper proposes a novel class of constraint-aware operators and an adaptive framework that improves quantum optimization efficiency and scalability for constrained problems.

Findings

Outperforms penalty-based QAOA in convergence speed and solution quality.

Requires roughly half the gates compared to traditional methods.

Successfully applied to finance and physics benchmark problems.

Abstract

Constrained combinatorial optimization with strict linear constraints underpins applications in drug discovery, power grids, logistics, and finance, yet remains computationally demanding for classical algorithms, especially at large scales. The Quantum Approximate Optimization Algorithm (QAOA) offers a promising quantum framework, but conventional penalty-based formulations distort optimization landscapes and demand deep circuits, undermining scalability on near-term hardware. In this work, we introduce Hamming Weight Operators, a new class of constraint-aware operators that confine quantum evolution strictly within the feasible subspace. Building on this idea, we develop Adaptive Hamming Weight Operator QAOA, which dynamically selects the most effective operators to construct shallow, problem-tailored circuits. We validate our approach on benchmark tasks from both finance and high-energy physics, specifically portfolio optimization and two-jet clustering with energy balance. Across these problems, our method inherently satisfies all constraints by construction, converges faster, and achieves higher Approximation Ratios than penalty-based QAOA, while requiring roughly half as many gates. By embedding constraint-aware operators into an adaptive variational framework, our approach establishes a scalable and hardware-efficient pathway for solving practical constrained optimization problems on near-term quantum devices.