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Quantum data learning of topological-to-ferromagnetic phase transitions in the 2+1D toric code loop gas model

Shamminuj Aktar1,*, Rishabh Bhardwaj1, Andreas Bärtschi1, Tanmoy Bhattacharya2, and Stephan Eidenbenz1

  • *Contact author: saktar@lanl.gov
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Phys. Rev. D 113, 114523 – Published 26 June, 2026

DOI: https://doi.org/10.1103/4d35-kw5d

Abstract

Quantum data learning (QDL) provides a framework for extracting physical insights directly from quantum states, bypassing the need for any identification of the classical observable of the theory. A central challenge in many-body physics is that the identity of quantum phases, especially those with topological order, is often inaccessible through local observables or simple symmetry-breaking diagnostics. Here, we apply QDL techniques to the 2+1-dimensional toric-code loop-gas model in a magnetic field. Ground states are generated across multiple lattice sizes using a parametrized loop-gas circuit with a variational quantum-eigensolver approach. We then train a quantum convolutional neural network across the full field-parameter range to perform phase classification and capture the overall phase structure. We also employ a physics-aware training protocol that excludes the near-critical region (0.2≤x≤0.4) around xc≈0.25, the phase-transition point estimated by quantum Monte Carlo simulation, reserving this window for testing to evaluate the ability of the model to learn the phase transition. In parallel, we implement an unsupervised quantum k-means method based on state overlaps, which partitions the dataset into two phases without prior labeling. Our supervised QDL approach recovers the phase structure and accurately locates the phase transition, in close agreement with previously reported values; the unsupervised QDL approach recovers the phase structure and locates the phase transition with a small offset as expected in finite volumes; both QDL methods outperform classical alternatives. These findings establish QDL as an effective framework for characterizing topological quantum matter, studying finite volume effects, and probing phase diagrams of higher-dimensional systems.

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Phys. Rev. D 113, 114523– Published 26 June, 2026
  • Received 11 December 2025
  • Accepted 8 May 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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