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.

翻译：受柯尔莫哥洛夫复杂度与无监督机器学习之间密切关系的启发，我们探索量子电路复杂度——量子计算与量子信息科学中的一个重要概念——作为理解和构建用于量子多体系统中拓扑序的可解释且高效的无监督机器学习的支点。我们认为，Nielsen的量子电路复杂度代表了拓扑量子多体物质相之间的内在拓扑距离，因此在拓扑序的可解释流形学习中起着核心作用。为了将概念力量转化为实际应用，我们提出了两个定理，分别将用于两个任意量子多体态之间量子路径规划的Nielsen量子电路复杂度与量子费希尔复杂度（Bures距离）和纠缠生成联系起来。利用这些联系，我们构建了基于保真度和基于纠缠的相似性度量或核函数，这些方法更便于实际实现。使用所提出的两种距离度量，我们对键交替XXZ自旋链的量子相、Kitaev环面码的基态以及随机乘积态进行了无监督流形学习，展示了其优越性能。此外，我们发现基于纠缠的方法能够捕捉拓扑序量子纠缠的长程结构，对局部Haar随机噪声具有更强的鲁棒性。我们还讨论了与经典影子层析成像和影子核学习的关系，其中后者可以从我们的方法中得到自然理解。我们的结果在量子电路计算、量子复杂度、量子计量学和拓扑量子序机器学习的关键概念与工具之间建立了联系。