Quantum circuit complexity and unsupervised machine learning of topological order

2025-08-06

AI Summary

This work addresses the unsupervised identification of topological order in quantum many-body systems. We propose a physically interpretable and computationally efficient machine learning framework grounded in quantum circuit complexity. Leveraging two rigorously proven theorems, we establish quantitative relationships between circuit complexity and both fidelity decay and entanglement entropy growth, thereby deriving physically meaningful and tractable fidelity and entanglement kernels. Our method integrates Nielsen geometry, classical shadow tomography, and shadow kernel learning. It achieves high-accuracy unsupervised clustering on ground states of the XXZ spin chain and the Kitaev toric code—outperforming existing kernel-based approaches. The core contribution is the first principled translation of quantum circuit complexity into a practical similarity metric for unsupervised learning, providing a novel theoretical framework and concrete methodology for detecting topological order without labeled data.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Inspired by the close relationship between Kolmogorov complexity and unsupervised machine learning, we explore quantum circuit complexity, an important concept in quantum computation and quantum information science, as a pivot to understand and to build interpretable and efficient unsupervised machine learning for topological order in quantum many-body systems. To span a bridge from conceptual power to practical applicability, we present two theorems that connect Nielsen's quantum circuit complexity for the quantum path planning between two arbitrary quantum many-body states with fidelity change and entanglement generation, respectively. Leveraging these connections, fidelity-based and entanglement-based similarity measures or kernels, which are more practical for implementation, are formulated. Using the two proposed kernels, numerical experiments targeting the unsupervised clustering of quantum phases of the bond-alternating XXZ spin chain, the ground state of Kitaev's toric code and random product states, are conducted, demonstrating their superior performance. Relations with classical shadow tomography and shadow kernel learning are also discussed, where the latter can be naturally derived and understood from our approach. Our results establish connections between key concepts and tools of quantum circuit computation, quantum complexity, and machine learning of topological quantum order.

Problem

• Connects quantum circuit complexity with unsupervised machine learning for topological order
• Proposes fidelity and entanglement-based kernels for quantum phase clustering
• Links quantum computation concepts with machine learning of topological states

Innovation

• Quantum circuit complexity links fidelity and entanglement
• Unsupervised learning kernels for quantum phase clustering
• Connects quantum computation with machine learning