Machine Learning Struggles to Grasp Complex States of Matter, Research Confirms

Scientists are increasingly exploring the intersection of quantum physics and machine learning, and a new study by Tarun Advaith Kumar, Yijian Zou, and Amir-Reza Negari, all from the Perimeter Institute for Theoretical Physics and University of Waterloo, working with colleagues Roger G. Melko and Timothy H. Hsieh also from the Perimeter Institute for Theoretical Physics and University of Waterloo, reveals fundamental limits to what machine learning algorithms can achieve. The researchers demonstrate that certain complex phases of matter, specifically those characterised by locally indistinguishable states, present a computational barrier to unsupervised learning techniques such as autoregressive neural networks. This work, which utilises conditional mutual information as a diagnostic tool and restricted statistical query models, establishes a link between the difficulty of learning a distribution and the presence of non-local correlations, potentially offering a new method for identifying exotic phases of matter and error-correction thresholds in quantum systems.

This work focuses on unsupervised learning, where algorithms discern patterns from unlabeled data, and reveals that autoregressive neural networks struggle to grasp the global properties of distributions characterised by locally indistinguishable (LI) states. These LI states, possessing identical local characteristics but differing globally, present a significant obstacle to accurate data representation. Researchers established a connection between conditional mutual information (CMI), a measure of statistical dependence, and the presence of LI states, finding that long-range CMI in a classical distribution indicates a spatially LI counterpart. By introducing a restricted statistical query model, the study proves that nontrivial phases exhibiting long-range CMI, such as those arising in strong-to-weak spontaneous symmetry breaking, are inherently hard to learn. This theoretical finding was validated through extensive simulations employing recurrent, convolutional, and Transformer neural networks trained on the syndrome and physical distributions of toric/surface codes subjected to bit flip noise. The results suggest that the difficulty of learning can serve as a diagnostic tool for identifying mixed-state phases, transitions, and even error-correction thresholds in physical systems. Furthermore, the research proposes that CMI, and more broadly “non-local Gibbsness”, can function as metrics for quantifying the inherent difficulty of learning a given distribution. This work bridges the gap between machine learning and condensed matter physics, offering a novel perspective on data analysis and potentially informing the development of more robust and reliable artificial intelligence. The identification of LI and CMI as key indicators of learning hardness could have significant implications for AI safety and the design of algorithms capable of handling complex, real-world data. Moreover, the concept of (un)learnability provides a new computational method for probing mixed-state phases and transitions, with potential applications in both numerical simulations and quantum experiments. The study centres on demonstrating computational hardness through the lens of mixed-state phases of matter and their learnability by autoregressive neural networks; a 72-qubit superconducting processor was not used. The research employed a restricted statistical query model to investigate the challenges of learning distributions characterised by locally indistinguishable (LI) states, a concept describing scenarios where two quantum states cannot be distinguished by local measurements. This approach allows for a rigorous theoretical analysis of learning complexity, moving beyond empirical observations of neural network performance. To generate training data, the team utilised Monte Carlo methods, producing datasets of 100,000 samples for most systems. Crucially, for each sample, the true probability was calculated using tensor network methods, providing an accurate benchmark for evaluating the neural networks’ performance. A notable exception was the 1D Transverse-Field Ising model, where a more efficient matrix-product state (MPS) technique, optimised with density matrix renormalization group (DMRG), was implemented for both sampling and probability calculation. For the surface code, a modification to the Monte Carlo samples was introduced, grouping spins to create local degrees of freedom of dimension four, and adding artificial spins at boundaries to maintain consistency. Optimisation of the neural networks involved a batch size of 128 and the Adabelief optimizer, initialised with a learning rate of 5×10−5. To ensure robust results, up to 30 copies of each network were trained with different random initializations for each parameter point, running for 400,000 steps. KL-divergence, a measure of how one probability distribution differs from another, was used to quantify performance, calculated using 100,000 samples distinct from the training set. All numerical experiments were implemented using JAX, a high-performance numerical computation library, and the NetKet library, which provides tools for quantum many-body physics. The Hilbert space, representing the state space of the quantum system, was constructed as a 1D hypercube graph, even for 2D systems, to facilitate autoregressive modelling. Initial analysis reveals that autoregressive neural networks struggle to learn distributions exhibiting locally indistinguishable (LI) states, a finding demonstrated through both theoretical proofs and extensive numerical experiments. Specifically, the study establishes polynomial-time lower bounds for learning within a restricted statistical query model, indicating a fundamental hardness associated with these types of distributions. The work demonstrates that conditional mutual information (CMI) serves as a diagnostic for LI, showing that long-range CMI in classical distributions implies the existence of a spatially LI partner. The research introduces a restricted statistical query learning model mirroring gradient-based training in autoregressive neural networks, and within this framework, locally indistinguishable states prove computationally intractable. Conversely, one-dimensional distributions with short-ranged CMI are efficiently learnable, aligning with existing results on Gibbs state learning. A toy model illustrates how LI states lead to vanishing gradients during the learning process, hindering effective training. Extensive numerical evidence supports these theoretical predictions across various architectures, recurrent neural networks (RNNs), convolutional neural networks (CNNs), and Transformers, and target systems, including syndrome and physical distributions of quantum and classical error-correcting codes with noise. The relentless pursuit of ever-more-powerful machine learning models often feels divorced from fundamental limits. This work offers a reminder that not everything can be learned, regardless of dataset size or architectural innovation. Researchers have identified a deep connection between the difficulty of learning certain kinds of probability distributions and the presence of “locally indistinguishable” states within them, essentially meaning the model cannot reliably tell these states apart. This isn’t merely a technical hurdle; it’s a statement about the inherent limitations of learning from data when the data itself contains irreducible ambiguity. For years, the more “non-local” the information within a dataset, the harder it will be for a machine to grasp its underlying structure. Pinpointing these limits is itself a challenge, and the diagnostic tools presented here, particularly conditional mutual information, are promising but require further refinement. Future research might explore whether these “hard to learn” distributions are systematically underrepresented in training datasets, or whether new learning paradigms are needed to overcome these fundamental barriers. Ultimately, understanding these limitations isn’t about abandoning machine learning, but about directing its power towards problems where success is genuinely attainable.