Quantum circuit complexity and unsupervised machine learning of topological order
arXiv:2508.04486 · doi:10.1038/s41467-026-71283-5
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. 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.
Updated version; With enriched Supplementary Information; 23 pages; 5 figures. Code is available upon reasonable request, and will be open-sourced along with the publication. Comments are welcome
References in corpus (74)
- Topological Insulators
- Topological insulators and superconductors
- Quantum Machine Learning
- The density-matrix renormalization group in the age of matrix product states
- Machine learning and the physical sciences
- Topological entanglement entropy
- Solving the Quantum Many-Body Problem with Artificial Neural Networks
- Detecting topological order in a ground state wave function
- Machine learning phases of matter
- Predicting Many Properties of a Quantum System from Very Few Measurements
- A class of quantum many-body states that can be efficiently simulated
- Complexity Equals Action
- Quantum spin squeezing
- Probing Topological Spin Liquids on a Programmable Quantum Simulator
- Complexity and Shock Wave Geometries
- Learning phase transitions by confusion
- Quantum Fisher information matrix and multiparameter estimation
- Lieb-Robinson bounds and the generation of correlations and topological quantum order
- Complexity, action, and black holes
- Quantum Computation as Geometry
- Quantum speed limit for physical processes
- Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy)
- Towards Complexity for Quantum Field Theory States
- Detection of Symmetry Protected Topological Phases in 1D
- Identifying topological order through unsupervised machine learning
- Quantum Loop Topography for Machine Learning
- Undecidability of the Spectral Gap (short version)
- Machine Learning Topological Invariants with Neural Networks
- Provably efficient machine learning for quantum many-body problems
- Identifying Quantum Phase Transitions using Artificial Neural Networks on Experimental Data
- Circuit Complexity in Fermionic Field Theory
- Circuit complexity for free fermions
- Unsupervised machine learning and band topology
- Linear growth of quantum circuit complexity
- Machine learning meets quantum physics
- Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems
- Models of quantum complexity growth
- Gapped quantum liquids and topological order, stochastic local transformations and emergence of unitarity
- Discriminative Cooperative Networks for Detecting Phase Transitions
- Unsupervised Manifold Clustering of Topological Phononics
- Diagnostics of mixed-state topological order and breakdown of quantum memory
- Entanglement rates and area laws
- Topological quantum phase transitions retrieved through unsupervised machine learning
- Many-body topological invariants from randomized measurements
- Quantum Information Meets Quantum Matter -- From Quantum Entanglement to Topological Phase in Many-Body Systems
- Deep Learning Topological Invariants of Band Insulators
- Characterization of topological states via dual multipartite entanglement
- Machine Learning Topological Phases with a Solid-state Quantum Simulator
- Upper bounds on entangling rates of bipartite Hamiltonians
- Unsupervised identification of topological order using predictive models
- Intrinsic Mixed-state Topological Order
- A Noisy Approach to Intrinsically Mixed-State Topological Order
- Entangling power and quantum circuit complexity
- Circuit Complexity across a Topological Phase Transition
- Post-Quench Evolution of Complexity and Entanglement in a Topological System
- A geometric approach to quantum circuit lower bounds
- Towards a classification of mixed-state topological orders in two dimensions
- Approximate Autonomous Quantum Error Correction with Reinforcement Learning
- Unsupervised Learning of Non-Hermitian Topological Phases
- Volume-law to area-law entanglement transition in a non-unitary periodic Gaussian circuit
- Unsupervised learning using topological data augmentation
- Quantum topology identification with deep neural networks and quantum walks
- Learning quantum states and unitaries of bounded gate complexity
- Nonanalyticity of circuit complexity across topological phase transitions
- Almost All Quantum States Have Low Entropy Rates for Any Coupling to the Environment
- Enhanced estimation of quantum properties with common randomized measurements
- Complexity and order in approximate quantum error-correcting codes
- Universality in long-distance geometry and quantum complexity
- Exponentially improved efficient machine learning for quantum many-body states with provable guarantees
- Efficient Learning for Linear Properties of Bounded-Gate Quantum Circuits
- Identifying topology of leaky photonic lattices with machine learning
- Exact Quantum Algorithms for Quantum Phase Recognition: Renormalization Group and Error Correction
- Efficient Learning of Long-Range and Equivariant Quantum Systems
- Provably Efficient Learning of Phases of Matter via Dissipative Evolutions