Mechanisms for Quantum Advantage in Global Optimization of Nonconvex Functions

2025-10-03

AI Summary

This work investigates quantum acceleration mechanisms for global optimization of nonconvex functions, moving beyond the conventional quantum tunneling paradigm. We propose a real-space adiabatic quantum algorithm that achieves polynomial-time convergence on potentials with non-degenerate global minima. By establishing a deep connection between the spectral properties of the Schrödinger operator and the mixing time of Langevin diffusion, we identify a novel source of quantum advantage: classical diffusion suffers exponential slowdown under WKB potentials with near-degenerate minima, whereas quantum evolution remains insensitive to barrier geometry. Leveraging semiclassical analysis, intrinsic ultracontractivity, and spectral theory, we prove polynomial convergence for broad nonconvex classes—including block-separable and perturbed strongly convex functions—while generic classical algorithms require exponential time. Numerical experiments confirm the predicted classical bottleneck and demonstrate significant quantum speedup.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

We present new theoretical mechanisms for quantum speedup in the global optimization of nonconvex functions, expanding the scope of quantum advantage beyond traditional tunneling-based explanations. As our main building-block, we demonstrate a rigorous correspondence between the spectral properties of Schrödinger operators and the mixing times of classical Langevin diffusion. This correspondence motivates a mechanism for separation on functions with unique global minimum: while quantum algorithms operate on the original potential, classical diffusions correspond to a Schrödinger operators with a WKB potential having nearly degenerate global minima. We formalize these ideas by proving that a real-space adiabatic quantum algorithm (RsAA) achieves provably polynomial-time optimization for broad families of nonconvex functions. First, for block-separable functions, we show that RsAA maintains polynomial runtime while known off-the-shelf algorithms require exponential time and structure-aware algorithms exhibit arbitrarily large polynomial runtimes. These results leverage novel non-asymptotic results in semiclassical analysis. Second, we use recent advances in the theory of intrinsic hypercontractivity to demonstrate polynomial runtimes for RsAA on appropriately perturbed strongly convex functions that lack global structure, while off-the-shelf algorithms remain exponentially bottlenecked. In contrast to prior works based on quantum tunneling, these separations do not depend on the geometry of barriers between local minima. Our theoretical claims about classical algorithm runtimes are supported by rigorous analysis and comprehensive numerical benchmarking. These findings establish a rigorous theoretical foundation for quantum advantage in continuous optimization and open new research directions connecting quantum algorithms, stochastic processes, and semiclassical analysis.

Problem

• Developing quantum mechanisms for global optimization of nonconvex functions
• Establishing quantum-classical separation without tunneling-based explanations
• Proving polynomial-time quantum optimization for broad nonconvex function families

Innovation

• Links Schrödinger operators to classical diffusion mixing
• Uses real-space adiabatic algorithm for nonconvex optimization
• Achieves polynomial runtime via semiclassical and hypercontractivity analysis