  Optimization problems are prevalent in various fields, and the gradient-based
gradient descent algorithm is a widely adopted optimization method. However, in
classical computing, computing the numerical gradient for a function with $d$
variables necessitates at least $d+1$ function evaluations, resulting in a
computational complexity of $O(d)$. As the number of variables increases, the
classical gradient estimation methods require substantial resources, ultimately
surpassing the capabilities of classical computers. Fortunately, leveraging the
principles of superposition and entanglement in quantum mechanics, quantum
computers can achieve genuine parallel computing, leading to exponential
acceleration over classical algorithms in some cases.In this paper, we propose
a novel quantum-based gradient calculation method that requires only a single
oracle calculation to obtain the numerical gradient result for a multivariate
function. The complexity of this algorithm is just $O(1)$. Building upon this
approach, we successfully implemented the quantum gradient descent algorithm
and applied it to the Variational Quantum Eigensolver (VQE), creating a pure
quantum variational optimization algorithm. Compared with classical
gradient-based optimization algorithm, this quantum optimization algorithm has
remarkable complexity advantages, providing an efficient solution to
optimization problems.The proposed quantum-based method shows promise in
enhancing the performance of optimization algorithms, highlighting the
potential of quantum computing in this field.
