Quantum circuit complexity and unsupervised machine learning of topological order

Enabling the discovery of unknown quantum many-body phases of matter remains a fundamental challenge in machine learning for quantum physics. Here, inspired by the close relationship between Kolmogorov complexity and unsupervised machine learning, we explore quantum circuit complexity as a pivot to build intuitive and efficient unsupervised machine learning for topological order in quantum many-body systems. We argue that Nielsen's quantum circuit complexity serves as an intrinsic informational distance between topological quantum states that results in interpretable manifold learning. To span a bridge from conceptual power to practical applicability, we present two theorems that connect Nielsen's quantum circuit complexity of quantum path planning with quantum Fisher complexity (Bures distance) and entanglement generation, respectively. The resulting kernel functions demonstrate superior performance and enhanced interpretability in numerical multiqubit experiments. Our results establish connections between key concepts of quantum computation, quantum complexity, quantum metrology, and machine learning of topological quantum order.