Grover-mixer Quantum Algorithm Achieves Superior Results for Higher-Order Optimization Problems

Quantum optimisation represents a major frontier in the pursuit of practical quantum computation, and researchers continually seek algorithms that can effectively tackle complex problems. Evgeniy Kiktenko, Elizaveta Krendeleva, and Aleksey Fedorov, all from the National University of Science and Technology “MISIS”, investigate a promising new approach to solving higher-order quadratic unconstrained optimisation problems, a challenging class of tasks with broad applications. Their work centres on the Grover-mixer variant of the Quantum Approximate Optimisation Algorithm, demonstrating that it consistently improves performance as computational steps increase, unlike more conventional methods. Crucially, the team also develops a theoretical framework to predict optimal settings for the algorithm, significantly reducing the computational burden and paving the way for implementation on existing quantum processors, thereby establishing a highly effective strategy for tackling difficult optimisation challenges.

GM-QAOA offers a compelling alternative due to its global search capabilities. This work investigates the application of GM-QAOA to Higher-Order Unconstrained Binary Optimization (HUBO) problems, also known as Polynomial Unconstrained Binary Optimization (PUBO), which represent a generalized class of combinatorial optimization tasks characterized by multi-variable interactions. The team presents a comprehensive numerical study demonstrating that GM-QAOA exhibits monotonic performance improvement with circuit depth and achieves superior results for HUBO problems, unlike XM-QAOA. A key innovation is the use of Extreme Value Theory (EVT) to analyze the distribution of solutions obtained by QAOA, allowing the authors to characterize performance, improve parameter tuning, and potentially mitigate noise effects. This approach also provides insights into how QAOA’s performance scales with problem size. The authors also explore the practical challenges of implementing QAOA on real quantum hardware and discuss potential strategies for overcoming these challenges. This work combines concepts from quantum computing, optimization, and statistics, demonstrating an interdisciplinary approach. The techniques developed could be applied to a wide range of optimization problems, including machine learning, finance, logistics, materials science, and drug discovery.

Grover-Mixer QAOA Excels at Optimization Problems

Scientists have achieved a significant breakthrough in quantum optimization by demonstrating the superior performance of a Grover-mixer based Quantum Approximate Optimization Algorithm (GM-QAOA) when applied to Higher-Order Unconstrained Binary Optimization (HUBO) problems. This work establishes GM-QAOA as a powerful tool for tackling complex combinatorial optimization tasks, differing substantially from traditional approaches utilizing transverse-field mixers (XM-QAOA). Experiments reveal that GM-QAOA exhibits monotonic performance improvement as circuit depth increases, a characteristic not observed with XM-QAOA, and consistently delivers superior results for HUBO problems. The team developed an analytical framework to model the dynamics of GM-QAOA, enabling the classical approximation of optimal parameters and significantly reducing the computational overhead associated with optimization.

This resource-efficient parameterized GM-QAOA nearly matches the performance of a fully optimized algorithm, while demanding considerably fewer computational resources. Measurements confirm that GM-QAOA dynamically couples all basis states, facilitating more effective exploration of complex energy landscapes and accelerating convergence towards optimal solutions. Further analysis demonstrates the inherent non-locality of the Grover mixer, potentially requiring highly entangling multi-qubit operations, yet the research suggests mitigation of hardware costs through implementation in emerging multi-level quantum platforms. The team demonstrated that, unlike the more traditional transverse-field mixer, the Grover-mixer exhibits consistently improving performance as the computational circuit deepens, ultimately achieving superior results for these challenging HUBO problems. Crucially, they developed an analytical framework to model the Grover-mixer’s behaviour, allowing for classical pre-optimization of parameters and a substantial reduction in the computational resources required. The findings reveal that a resource-efficient, analytically optimized Grover-mixer approach closely matches the performance of a fully optimized algorithm, while demanding far less computational power, establishing a practical route towards implementing this method on existing quantum hardware.

Results indicate that the performance of this optimized Grover-mixer improves markedly with increasing problem complexity, even surpassing the performance of the transverse-field mixer in certain scenarios, particularly when dealing with higher-order interactions within the optimization problem. Future work could focus on exploring the interplay between analytical parameter selection and problem structure to further enhance performance and scalability. The team also notes that observed trends in critical circuit depth, where the Grover-mixer outperforms the transverse-field mixer, warrant further investigation to fully understand the relationship between problem size, interaction order, and algorithm efficiency. These results represent a valuable step towards harnessing the potential of near-term quantum processors for tackling real-world optimization challenges.