Quantum circuit complexity and unsupervised machine learning of topological order
This work addresses the challenge of interpretable and efficient unsupervised learning for topological quantum order, which is important for quantum computation and information science, though it appears incremental by building on existing concepts.
The authors tackled the problem of unsupervised machine learning for topological order in quantum many-body systems by connecting quantum circuit complexity to practical similarity measures, demonstrating superior clustering performance on quantum phases like the bond-alternating XXZ spin chain and Kitaev's toric code.
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.