In this paper, we identify a family of nonconvex continuous optimization instances, each $d$-dimensional instance with $2^d$ local minima, to demonstrate a quantum-classical performance separation. Specifically, we prove that the recently proposed Quantum Hamiltonian Descent (QHD) algorithm [Leng et al., arXiv:2303.01471] is able to solve any $d$-dimensional instance from this family using $\widetilde{\mathcal{O}}(d^3)$ quantum queries to the function value and $\widetilde{\mathcal{O}}(d^4)$ additional 1-qubit and 2-qubit elementary quantum gates. On the other side, a comprehensive empirical study suggests that representative state-of-the-art classical optimization algorithms/solvers (including Gurobi) would require a super-polynomial time to solve such optimization instances.

翻译：本文识别了一族非凸连续优化实例，每个$d$维实例包含$2^d$个局部最小值，用以展示量子与经典算法间的性能分离。具体而言，我们证明了近期提出的量子哈密顿下降算法[Leng et al., arXiv:2303.01471]能够通过$\widetilde{\mathcal{O}}(d^3)$次函数值量子查询和$\widetilde{\mathcal{O}}(d^4)$个额外的单量子比特和双量子比特基本量子门，求解该族中任意$d$维实例。另一方面，全面的实证研究表明，代表性最先进的经典优化算法/求解器（包括Gurobi）需要超多项式时间才能求解此类优化实例。