TetrisCNN for interpretable detection of phases of matter from experimental quantum simulator data

Technical Summary: TetrisCNN for Interpretable Detection of Phases of Matter

Problem Statement
Detecting phases of matter in complex quantum systems, particularly those with long-range interactions and strong entanglement, remains a central challenge. Traditional methods rely on identifying correct order parameters, a task often dependent on expert intuition and educated guesses, which becomes intractable for unknown transitions. While neural networks have offered alternatives for locating phase transitions without a priori physical knowledge, they typically function as "black boxes." They identify phases without elucidating their underlying properties or providing interpretable descriptions. Furthermore, these models often struggle with realistic, noisy experimental data, where noise can act as adversarial perturbation, and the "Rashomon effect" (the existence of multiple equally predictive but qualitatively distinct models) undermines the stability of inferred conclusions. The ultimate goal is to develop automated approaches that are not only accurate but also interpretable, enabling the discovery of new phases and the understanding of their physical properties, specifically the underlying order parameters.

Methodology: TetrisCNN Architecture
The authors introduce TetrisCNN, a convolutional neural network architecture designed to learn sparse, interpretable latent representations directly in terms of spin correlators. The architecture is motivated by the need to provide an explicit drive to "compress knowledge."

• Parallel Branches with Diverse Filters: The network consists of parallel convolutional branches, each utilizing filters of different shapes and sizes (reminiscent of Tetris blocks). These filters represent candidate local correlators at various spatial scales and geometries.
• Boolean-Fourier Mapping: A key theoretical component is the mapping of convolutional operations on Boolean-valued spin data to spin correlators. By combining local convolutional functions with Global Average Pooling (GAP), the output of any branch can be expressed as a linear combination of spin correlators $C[P']$ (where $P'$ are subpatterns of the filter pattern $P$). This is derived via Boolean Fourier expansion.
• Sparsity Regularization: To address redundancy and the Rashomon effect, the authors employ L1-regularization on the branch outputs (the "bottleneck"). This penalizes the magnitude of branch activations, with a specific penalty structure ($\lambda_k$) that increases with the complexity (cardinality) of the filter pattern. This forces the network to automatically select a minimal subset of correlators sufficient for the learning task, suppressing irrelevant or redundant branches.
• Rotational Invariance: To respect the $C_4$ symmetry of the square lattice, filters are applied in all distinct 90° rotations, and the resulting feature maps are averaged, ensuring the output is rotationally invariant.
• Task Network: The sparse latent variables (branch activations) are fed into a standard fully connected "task network" (classifier). Because the latent space is low-dimensional and physically interpretable, the decision boundary of this classifier can be approximated by symbolic formulas (e.g., polynomials) in terms of the spin correlators.

Experimental Data and Tasks
The method is applied to raw projective measurement snapshots (experimental data) from two Rydberg-atom quantum simulators:

1. 2D Ising Model (Transverse-Field Ising Model): An $8 \times 8$ array. Snapshots are measured in the Z-basis. The system undergoes a protocol involving a crossover from a Z-polarized paramagnetic state and a subsequent transition to an antiferromagnetic phase.
2. XY Model: A $6 \times 7$ rectangular array. Snapshots are measured in both X and Z bases. The protocol involves an adiabatic ramp connecting a Néel state to a ferromagnetic state in the XY plane.

The learning task is formulated as supervised classification, where snapshots are labeled based on a transition point estimated by various unsupervised methods (Learning by Confusion, Prediction-Divergence, PCA, UMAP, Distance Learning).

Key Results

• Sparse Descriptor Selection: TetrisCNN successfully identifies a minimal set of active branches required for classification.
  • For the Ising dataset, only a single branch corresponding to a $1 \times 1$ filter ($z[\blacksquare]$) remains active. This branch maps directly to the Z-basis magnetization ($C_Z[\blacksquare]$).
  • For the XY dataset, three branches remain active: $z[\blacksquare]$, $z[\blacksquare\blacksquare]$, and $z[\blacksquare\square\blacksquare]$. These map to the X-basis magnetization ($C_X[\blacksquare]$), a rotationally invariant nearest-neighbor Z-correlator ($C_Z^{rot}[\blacksquare\blacksquare]$), and a diagonal Z-correlator ($C_Z^{rot}[\blacksquare\square\blacksquare]$).
• Symbolic Decision Boundaries: The decision boundaries in the latent space are expressed as symbolic formulas:
  • Ising: A simple threshold on Z-magnetization ($C_Z[\blacksquare] \approx 0.387$).
  • XY: A quadratic surface involving the squared X-magnetization and the identified Z-correlators. The quadratic term $C_X[\blacksquare]^2$ closely relates to the in-plane ferromagnetic order parameter $(m^X)^2_c$.
• Interpretation of Crossovers vs. Transitions: In the Ising data, the network detects the crossover out of the polarized state (driven by Z-magnetization loss) rather than the later antiferromagnetic phase transition, which is consistent with the smooth nature of the crossover in finite-size, noisy experimental data.
• Robustness: The identified branches and decision rules are robust to random initializations and variations in data pairing.

Significance and Claims
The paper claims that TetrisCNN bridges the gap between high-performance machine learning and physical interpretability. Its primary contributions are:

1. Interpretability by Design: Unlike post-hoc analysis methods, TetrisCNN's architecture inherently produces latent representations that are linear combinations of physically measurable spin correlators.
2. Handling Experimental Noise: The method is demonstrated on raw, noisy experimental data, showing it can extract meaningful physical insights despite finite-size effects, decoherence, and measurement errors.
3. Revealing Task-Dependent Physics: The authors emphasize that the identified observables are those most useful for the specific learning task (classification), which may or may not coincide with conventional order parameters. For instance, in the XY case, the network combines the physically expected squared X-magnetization with Z-basis correlators that provide stronger single-snapshot discrimination, a nuance that a black-box model would obscure.
4. Model Selection: The sparsity regularization acts as a model-selection mechanism, navigating the Rashomon effect to find the simplest, most interpretable solution among many functionally equivalent models.

The authors position TetrisCNN as a step toward "machine-assisted discovery," where interpretable networks can provide sparse representations of complex quantum data, facilitating a dialogue between experiment, theory, and data-driven models. They note limitations, such as the current implementation's focus on local correlators and the need to extend the branch dictionary to capture nonlocal or string-like correlations for more exotic phases.