Quantum-Inspired Hamiltonian Descent for Mixed-Integer Quadratic Programming

Large-scale non-convex optimization problems, often involving mixed-integer variables, arise naturally in domains such as finance and medical imaging. Conventional CPU-based solvers rely on sequential decision-making, which limits their ability to leverage modern hardware accelerators like GPUs. Motivated by recent advances in quantum optimization, particularly Quantum Hamiltonian Descent (QHD), we introduce Quantum-Inspired Hamiltonian Descent (QIHD), a family of algorithms designed for efficient large-scale optimization on GPU clusters. QIHD reformulates optimization tasks as classical dynamical systems and exploits massively parallel GPU simulations to accelerate computation. Despite being classical, QIHD inherits distinctive QHD-like properties—such as tunneling—and demonstrates similar empirical behavior. We provide a scalable GPU implementation in JAX for mixed-integer quadratic programming (MIQP) problems, capable of handling millions of nonzero elements within seconds. Extensive benchmarks on large-scale MIQP problems show that QIHD consistently outperforms the state-of-the-art CPU-based solver Gurobi when time budgets are limited, highlighting its potential as a practical and scalable optimization framework.