Jiaqi Leng: Quantum Dynamics for Continuous Optimization

Continuous optimization problems arise in virtually all disciplines of quantitative research, including applied mathematics, computer science, and operations research. Many real-world optimization problems exhibit sophisticated landscapes characterized by multiple local minima, rendering them intractable both in theory and practice. This talk focuses on Quantum Hamiltonian Descent (QHD), a quantum algorithm for continuous optimization. QHD is derived as the path integral of standard gradient descent (GD). It inherits the algorithmic simplicity of GD while exhibiting drastically different behavior due to the quantum interference of classical paths, particularly in nonlinear and nonconvex optimization. Specifically, we prove that QHD can efficiently solve a family of degree-4 polynomial optimization instances, each characterized by exponentially many local minima. Beyond the standard circuit-based implementation, we also introduce an analog implementation of QHD through the Hamiltonian embedding technique for sparse Hamiltonian simulation. Based on this approach, we developed an open-source software package named QHDOPT, which was used in an empirical study to confirm the practical advantages of QHD for large-scale nonlinear optimization problems.