Operator-Level Quantum Acceleration of Non-Logconcave Sampling

2025-05-08

AI Summary

Efficient sampling from non-logconcave Gibbs distributions $sigma propto e^{-eta V}$—where the potential $V$ is nonconvex—remains challenging, as classical Langevin dynamics suffer from slow convergence. Method: We propose the first operator-level quantum acceleration framework: encoding the Gibbs distribution into quantum state amplitudes, constructing continuous-time quantum sampling dynamics via block decomposition of the Witten Laplacian, and integrating singular value thresholding with quantum simulation to enable efficient sampling under nonconvex potentials. Results: Theoretically, we establish the first rigorous quantum advantage guarantee for non-logconcave settings. Algorithmically, we achieve the first quantum speedups for both standard Langevin dynamics and replica-exchange Langevin diffusion. Practically, we obtain the first provable quantum acceleration dependent on the Poincaré constant, significantly improving sampling efficiency and convergence rates in multimodal energy landscapes.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Large-scale security measurements

Abstract

Sampling from probability distributions of the form $sigma propto e^{-eta V}$, where $V$ is a continuous potential, is a fundamental task across physics, chemistry, biology, computer science, and statistics. However, when $V$ is non-convex, the resulting distribution becomes non-logconcave, and classical methods such as Langevin dynamics often exhibit poor performance. We introduce the first quantum algorithm that provably accelerates a broad class of continuous-time sampling dynamics. For Langevin dynamics, our method encodes the target Gibbs measure into the amplitudes of a quantum state, identified as the kernel of a block matrix derived from a factorization of the Witten Laplacian operator. This connection enables Gibbs sampling via singular value thresholding and yields the first provable quantum advantage with respect to the Poincar'e constant in the non-logconcave setting. Building on this framework, we further develop the first quantum algorithm that accelerates replica exchange Langevin diffusion, a widely used method for sampling from complex, rugged energy landscapes.

Problem

• Quantum acceleration for non-logconcave sampling dynamics
• Overcoming poor classical performance in non-convex potentials
• Enabling Gibbs sampling via quantum state encoding

Innovation

• Quantum algorithm accelerates non-logconcave sampling
• Encodes Gibbs measure into quantum state amplitudes
• Accelerates replica exchange Langevin diffusion