The gradient descent method is central to numerical optimization and is the key ingredient in many machine learning algorithms. It promises to find a local minimum of a function by iteratively moving along the direction of the steepest descent. Since for high-dimensional problems the required computational resources can be prohibitive, it is desirable to investigate quantum versions of the gradient descent, such as the recently proposed (Rebentrost et al.¹). Here, we develop this protocol and implement it on a quantum processor with limited resources. A prototypical experiment is shown with a four-qubit nuclear magnetic resonance quantum processor, which demonstrates the iterative optimization process. Experimentally, the final point converged to the local minimum with a fidelity >94%, quantified via full-state tomography. Moreover, our method can be employed to a multidimensional scaling problem, showing the potential to outperform its classical counterparts. Considering the ongoing efforts in quantum information and data science, our work may provide a faster approach to solving high-dimensional optimization problems and a subroutine for future practical quantum computers.

梯度下降法是数值优化的核心，并且是许多机器学习算法中的关键要素。它有望通过沿最陡下降方向反复移动来找到函数的局部最小值。由于对于高维问题，所需的计算资源可能是禁止的，因此希望研究梯度下降的量子形式，例如最近提出的（Rebentrost等人¹）。在这里，我们开发此协议并将其在资源有限的量子处理器上实现。显示了具有四量子位核磁共振量子处理器的原型实验，该实验演示了迭代优化过程。在实验上，最终点以全状态断层扫描量化的保真度> 94％收敛到局部最小值。此外，我们的方法可用于多维比例尺缩放问题，显示出超越其经典对应方法的潜力。考虑到量子信息和数据科学领域的不断努力，我们的工作可能会为解决高维优化问题提供更快的方法，并为将来的实用量子计算机提供子程序。