Quantum Optimisation Algorithm Surpasses Classical Methods with Enhanced Gradient Descent

The pursuit of efficient optimisation algorithms remains central to advances in machine learning and computational problem-solving. Researchers continually seek methods to navigate complex computational landscapes, avoiding local minima and converging rapidly on global solutions. A new approach, detailed in ‘Quantum Optimisation via Gradient-Based Hamiltonian Descent’, builds upon the principles of Hamiltonian dynamics – a framework describing the time evolution of a physical system – and incorporates gradient information to enhance performance. Jiaqi Leng (institutions 1 & 2) and Bin Shi (institutions 3 & 4) present a modified quantum Hamiltonian descent algorithm that demonstrably accelerates convergence and improves solution accuracy across a range of challenging optimisation problems, exceeding the performance of established classical methods by a significant margin.

Researchers have developed a novel quantum optimisation algorithm, Quantum Hamiltonian Descent (QHD), which exhibits improved performance compared with established classical techniques across a range of challenging problems. This approach leverages quantum tunnelling to navigate complex solution spaces, addressing limitations inherent in traditional optimisation methods. Numerical experiments confirm QHD’s superior performance on benchmark functions when compared with Stochastic Gradient Descent with Momentum (SGDM) and Nesterov Accelerated Gradient (NAG), representing an advancement in the field.

The core innovation of QHD lies in its ability to escape local minima and saddle points – common pitfalls that trap classical algorithms. Quantum tunnelling, a quantum mechanical phenomenon where a particle penetrates an energy barrier despite lacking the classical energy to do so, allows the algorithm to explore a wider range of potential solutions and identify the global optimum with greater efficiency. This is particularly valuable for non-convex problems, where the solution landscape is characterised by numerous local optima and gradient-based methods often fail to converge.

A key development detailed in the work is gradient-based QHD, a refined version incorporating gradient information into the optimisation process. This addresses limitations in convergence speed and robustness observed in highly non-convex problems, further improving its ability to find optimal solutions. By integrating gradient data, the algorithm exhibits accelerated convergence and a markedly increased probability of identifying global solutions.

The researchers establish a rigorous mathematical foundation for QHD, detailing the Hamiltonian, evolution equations, and gradient descent steps that govern its behaviour. The Hamiltonian, representing the total energy of the system, dictates its evolution over time. The evolution equations describe how the algorithm explores the solution space and refines its search strategy.

Comprehensive numerical analysis, utilising functions such as Styblinski-Tang, Michalewicz, Cube-Wave, and Rastrigin, provides a thorough evaluation of the algorithm’s capabilities. These benchmark functions represent a diverse range of challenges and allow for a fair comparison with other algorithms. Results demonstrate that gradient-based QHD consistently outperforms both existing quantum and classical methods, achieving a significant improvement in performance. Simulations reveal that gradient-based QHD outperforms existing methods by at least one order of magnitude.

The development of QHD represents a step forward in quantum optimisation, offering a promising new approach to solving challenging problems in a wide range of disciplines.

Future work will focus on scaling QHD to address larger, more complex problems. Applying the algorithm to real-world datasets and complex systems will require significant computational resources and algorithmic optimisations. Researchers plan to explore parallel computing techniques and distributed algorithms to accelerate computation and enable the algorithm to handle larger problem instances.

It is also crucial to consider the challenges associated with implementing QHD on current and near-term quantum hardware, including decoherence and gate errors. Researchers plan to investigate error mitigation techniques and develop algorithms that are robust to these limitations. A comparative analysis with other prominent quantum optimisation algorithms, such as Quantum Approximate Optimisation Algorithm (QAOA) and Variational Quantum Eigensolver (VQE), will provide valuable insights into the strengths and weaknesses of each approach.

Researchers plan to investigate QHD’s theoretical properties, including its convergence rate and ability to find the global optimum. Finally, exploring the potential of combining QHD with other optimisation techniques, such as classical heuristics and metaheuristics, could lead to even more powerful and efficient algorithms.