Quantum circuit complexity and unsupervised machine learning of topological order

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. We argue that Nielsen's quantum circuit complexity represents an intrinsic topological distance between topological quantum many-body phases of matter, and as such plays a central role in interpretable manifold learning of topological order. 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 quantum Fisher complexity (Bures distance) 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 distance measures, unsupervised manifold learning of quantum phases of the bond-alternating XXZ spin chain, the ground state of Kitaev's toric code and random product states, is conducted, demonstrating their superior performance. Moreover, we find that the entanglement-based approach, which captures the long-range structure of quantum entanglement of topological orders, is more robust to local Haar random noises. Relations with classical shadow tomography and shadow kernel learning are also discussed, where the latter can be naturally understood from our approach. Our results establish connections between key concepts and tools of quantum circuit computation, quantum complexity, quantum metrology, and machine learning of topological quantum order.