Hamming Weight Operators Advance QAOA Performance in Constrained Combinatorial Optimization at Large Scales

Constrained combinatorial optimisation presents a significant challenge for classical computers across diverse fields including drug discovery, logistics and financial modelling, particularly as problem scales increase. Yajie Hao, Qiming Ding, Xiao Yuan and colleagues from the University of Electronic Science and Technology of China and Peking University have now proposed a novel approach to tackle this issue using quantum computing. Their research introduces Hamming Weight Operators, a new type of constraint-aware operator designed to restrict quantum processing to only feasible solutions, avoiding the distortions inherent in traditional penalty-based methods. This innovation allows for the development of shallower, more efficient quantum circuits, demonstrably improving performance on benchmark problems in finance and high-energy physics with fewer computational gates. By constructing solutions that inherently satisfy constraints, this work paves the way for scalable and practical quantum optimisation on emerging hardware.

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. These operators directly enforce feasibility within the quantum Hilbert space, avoiding the landscape distortion inherent in penalty methods.

Adaptive QAOA with Hamming Weight Operators

The researchers tackled the challenge of constrained combinatorial optimization, a problem vital to fields like finance and high-energy physics, by pioneering a new approach to quantum computing. Their work centres on the Quantum Approximate Optimization Algorithm (QAOA), but directly addresses limitations inherent in traditional penalty-based constraint handling. Instead of adding penalty terms that distort the optimization landscape, the team engineered a novel class of operators termed Hamming Weight Operators. These operators fundamentally confine quantum evolution strictly within the feasible solution subspace, ensuring all constraints are inherently satisfied throughout the optimization process.

Building upon this innovation, scientists developed Adaptive Hamming Weight Operator QAOA (AHWO-QAOA), a dynamic system that intelligently selects the most effective operators for constructing shallow quantum circuits. Inspired by the ADAPT-VQE framework, AHWO-QAOA iteratively builds a problem-tailored ansatz, prioritizing both constraint satisfaction and minimal resource usage. Experiments employed numerical simulations on benchmark portfolio optimization problems, scaling up to 20 qubits, and also investigated two-jet clustering with energy balance, a complex task from high-energy physics. The study meticulously validated AHWO-QAOA’s performance against conventional penalty-based QAOA, revealing significant advantages.

The new method consistently achieved higher Approximation Ratios while requiring approximately half the number of quantum gates. Crucially, AHWO-QAOA demonstrated faster convergence, obtaining high-quality solutions in fewer iterations, and guaranteed feasibility by construction, a direct result of the constraint-aware operators. This approach establishes a scalable and hardware-efficient pathway for solving practical constrained optimization problems on near-term quantum devices, overcoming limitations of existing methods. The team’s technique reveals a substantial improvement in both the speed and accuracy of solving complex optimization problems. By directly embedding constraint awareness into the quantum circuit design, they circumvent the need for delicate parameter tuning and avoid the creation of rugged energy landscapes that plague penalty-based methods. This innovative methodology not only enhances performance on established benchmarks but also paves the way for tackling larger, more complex problems with limited quantum resources.

Hamming Operators Solve Constrained Quantum Optimisation

Scientists have achieved a breakthrough in constrained combinatorial optimization, a field underpinning applications in areas such as drug discovery, power grids, logistics, and finance. This innovative approach bypasses the limitations of conventional penalty-based formulations, which often distort optimization landscapes and necessitate complex quantum circuits. The team successfully developed Adaptive Hamming Weight Operator QAOA, a method that dynamically selects the most effective operators to construct shallow, problem-tailored circuits.

Experiments revealed that this new method inherently satisfies all linear constraints by construction, a significant advantage over penalty-based Quantum Approximate Optimization Algorithm (QAOA). Tests conducted on benchmark tasks, including portfolio optimization and two-jet clustering with energy balance from high-energy physics, demonstrate accelerated convergence and higher Approximation Ratios. Specifically, the Adaptive Hamming Weight Operator QAOA (AHWO-QAOA) achieved high-quality solutions in fewer iterations than traditional methods, while simultaneously reducing the number of elementary gates required by approximately half. These results were obtained on problems scaling up to 20 qubits, validating the approach’s potential for handling complex optimization challenges.

Measurements confirm that AHWO-QAOA delivers guaranteed feasibility, ensuring all linear constraints are consistently met throughout the optimization process. Data shows a marked improvement in convergence speed, with solutions obtained significantly faster compared to penalty-based QAOA. The breakthrough delivers a substantial reduction in resource requirements, utilizing nearly 50% fewer elementary gates while maintaining, and often exceeding, the accuracy of existing methods. The research team validated the performance of their framework across both financial and physics-inspired benchmarks, establishing a foundation for scalable constrained optimization on near-term quantum devices.

Adaptive Hamming Operators for Constrained Optimisation

This work introduces the Adaptive Hamming Weight Operator Quantum Approximate Optimization Algorithm, a new framework for tackling constrained combinatorial optimization. Numerical simulations, conducted on problems from finance and high-energy physics, demonstrate that this approach consistently satisfies constraints, converges more rapidly, and achieves superior Approximation Ratios with significantly shallower quantum circuits. The significance of this research lies in its potential to advance practical quantum optimization for real-world applications.

The Hamming Weight Operator itself is presented as a versatile building block, adaptable for use in other variational algorithms and quantum machine learning models where maintaining specific properties is essential. While the authors acknowledge limitations in the current scope, specifically focusing on linear constraints, they propose future research directions including extending the framework to handle inequality constraints and exploring more complex symmetries. This work establishes a scalable and hardware-efficient pathway towards solving constrained optimization problems on near-term quantum devices, offering both theoretical and practical benefits to the field.