Spectral Filtering for Learning Quantum Dynamics

2026-01-29

AI Summary

This work addresses the challenge of learning high-dimensional quantum systems, which is typically hindered by the curse of dimensionality due to the exponential growth in computational complexity associated with full density matrix reconstruction. The authors propose a quantum spectral filtering approach that models quantum linear response dynamics as a complex-valued linear dynamical system with sector-bounded eigenvalues. By introducing Slepian basis expansions within a non-orthodox dynamic learning framework, they establish— for the first time—that learnability depends solely on the effective quantum dimension rather than the ambient Hilbert space dimension. Integrating spectral bandwidth analysis with structured state-space models, the method achieves sample and computational complexities decoupled from the environmental dimension, thereby significantly mitigating the curse of dimensionality while preserving estimation accuracy.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Intelligent Robots: State Estimation

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

Learning high-dimensional quantum systems is a fundamental challenge that notoriously suffers from the curse of dimensionality. We formulate the task of predicting quantum evolution in the linear response regime as a specific instance of learning a Complex-Valued Linear Dynamical System (CLDS) with sector-bounded eigenvalues -- a setting that also encompasses modern Structured State Space Models (SSMs). While traditional system identification attempts to reconstruct full system matrices (incurring exponential cost in the Hilbert dimension), we propose Quantum Spectral Filtering, a method that shifts the goal to improper dynamic learning. Leveraging the optimal concentration properties of the Slepian basis, we prove that the learnability of such systems is governed strictly by an effective quantum dimension $k^*$, determined by the spectral bandwidth and memory horizon. This result establishes that complex-valued LDSs can be learned with sample and computational complexity independent of the ambient state dimension, provided their spectrum is bounded.

Problem

• quantum dynamics
• curse of dimensionality
• linear response
• high-dimensional quantum systems
• system identification

Innovation

• Quantum Spectral Filtering
• Complex-Valued Linear Dynamical System
• Slepian basis
• effective quantum dimension
• Structured State Space Models