Quantum Data Learning of Topological-to-Ferromagnetic Phase Transitions in the 2+1D Toric Code Loop Gas Model

TL;DR

This work demonstrates that quantum data learning (QDL) can identify a topological-to-ferromagnetic phase transition directly from ground-state quantum data in the 2+1D toric-code loop-gas model. By generating states with a variational loop-gas circuit and applying both supervised quantum convolutional neural networks (QCNN) and unsupervised quantum k-means, the study extracts phase structure and locates the critical point with finite-size scaling that agrees with quantum Monte Carlo benchmarks. QCNN achieves near-perfect phase classification and extrapolates to $x_c(\infty) = 0.2518 \pm 0.0260$, closely matching the known $x_c \approx 0.25$, while quantum $k$-means provides a competitive unsupervised estimate with a small offset. Classical ML baselines underperform compared to QCNN in identifying the phase boundary, highlighting the advantage of learning directly from quantum data for probing topological quantum matter and finite-size effects.

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, are 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 (PLGC) with a variational quantum-eigensolver (VQE) approach. We then train a quantum convolutional neural network (QCNN) 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 (x_c = 0.25), the phase-transition point estimated by quantum Monte Carlo, 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.

Figures (12)

• Figure 1: Overview of the workflow for quantum data learning of phases in the $2{+}1$-dimensional toric-code loop-gas model under a magnetic field. Ground states of $H{(x)}$ are approximated by a variational quantum eigensolver (VQE) using the parametrized loop-gas circuit (PLGC) on finite lattices. The resulting states $|\psi_x\rangle$ are processed using QDL techniques to learn the phase structure and distinguish between topologically ordered and ferromagnetically ordered phases.
• Figure 2: Toric geometry and boundary effects in the toric code model (TCM). (left) Construction of the toric geometry under periodic boundary conditions, yielding a manifold with two non-contractible Wilson loops $W_x$ and $W_y$ that define distinct topological sectors of the ground state and encode the intrinsic $\mathbb{Z}_2$ topological order. (right) Under open boundary conditions, the finite square lattice is topologically equivalent to a disk $D^2$, eliminating non-contractible loops and hence the global ground-state degeneracy while preserving the bulk $\mathbb{Z}_2$ order.
• Figure 3: QCNN evaluation with randomly split datasets. Each panel corresponds to a different lattice size: $2{\times}2$, $2{\times}3$, $3{\times}3$, and $4{\times}3$. Predicted phase labels closely follow the true labels for all lattice sizes, demonstrating near-perfect classification accuracy across system sizes.
• Figure 4: QCNN phase-boundary evaluation across lattice sizes. Raw QCNN outputs $\langle Z \rangle$ are shown as functions of the field parameter $x$ for lattices $2{\times}2$, $2{\times}3$, $3{\times}3$, and $4{\times}3$. The yellow-shaded regions indicate the phase flip-interval estimates of the transition point $x_c'\!\pm\!\Delta x$ extracted from the QCNN decision boundary, while the red dashed lines mark the QMC benchmark $x_c \approx 0.25$. As the system size increases, the QCNN-estimated phase boundary becomes close to the infinite-volume QMC critical point.
• Figure 5: Quantum $k$-means phase classification of toric-code ground states. Each panel corresponds to a different lattice size: $2{\times}2$, $2{\times}3$, $3{\times}3$, and $4{\times}3$. The shaded yellow regions denote the phase transition intervals $\hat{x}_c' \pm \Delta x$, while the red dashed line marks the QMC reference $x_c \approx 0.25$. The cluster assignments ($+1$ and $-1$) (jittered for visibility) exhibit clear phase separation and finite-size effects similar to the QCNN results shown in Fig. \ref{['fig:qcnn_physics_aware_raw_test']}.