Unlearnable phases of matter

2026-02-11

AI Summary

This work investigates the fundamental limitations of unsupervised machine learning in identifying mixed-state quantum phases that are locally indistinguishable yet possess global characteristics. By analyzing the failure mechanisms of autoregressive neural networks in capturing long-range conditional mutual information (CMI) distributions, the study establishes CMI as an effective diagnostic for local indistinguishability and proves the computational hardness of learning such phases within the statistical query model. Innovatively, it proposes learning hardness itself as a novel probe for detecting mixed-state phases, phase transitions, and quantum error correction thresholds, introducing “non-local Gibbsianness” and CMI as quantifiable measures of learnability. The theoretical predictions are validated through experiments employing RNN, CNN, and Transformer architectures on systems exhibiting strong and weak spontaneous symmetry breaking, as well as topological codes—such as the surface code—under bit-flip noise.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Search and Optimization: Learning to Search

Application Category

• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We identify fundamental limitations in machine learning by demonstrating that non-trivial mixed-state phases of matter are computationally hard to learn. Focusing on unsupervised learning of distributions, we show that autoregressive neural networks fail to learn global properties of distributions characterized by locally indistinguishable (LI) states. We demonstrate that conditional mutual information (CMI) is a useful diagnostic for LI: we show that for classical distributions, long-range CMI of a state implies a spatially LI partner. By introducing a restricted statistical query model, we prove that nontrivial phases with long-range CMI, such as strong-to-weak spontaneous symmetry breaking phases, are hard to learn. We validate our claims by using recurrent, convolutional, and Transformer neural networks to learn the syndrome and physical distributions of toric/surface code under bit flip noise. Our findings suggest hardness of learning as a diagnostic tool for detecting mixed-state phases and transitions and error-correction thresholds, and they suggest CMI and more generally ``non-local Gibbsness''as metrics for how hard a distribution is to learn.

Problem

• unlearnable phases
• mixed-state phases
• locally indistinguishable states
• conditional mutual information
• statistical query model

Innovation

• unlearnable phases
• conditional mutual information
• locally indistinguishable states
• statistical query model
• non-local Gibbsness