Unsupervised Quantum Phase Detection

• Unsupervised quantum phase detection is a method that uses clustering and dimensionality reduction on raw eigenstate data to reveal distinct quantum phases and transitions.
• It employs algorithms like DBSCAN and OPTICS to differentiate between extended, localized, and critical states, achieving up to 98.4% similarity with traditional IPR diagnostics.
• This approach is scalable to many-body systems and robust against noise, enabling automatic phase diagram extraction without relying on pre-labeled data.

Unsupervised quantum phase detection encompasses a spectrum of machine learning and data-analytic methodologies designed to identify and characterize distinct phases and phase transitions in quantum many-body systems without reliance on a priori knowledge of order parameters or manually labeled training data. These approaches operate through the statistical analysis, clustering, and dimensionality reduction of high-dimensional data extracted from theoretical simulations or experimental measurements, aiming to autonomously uncover physically meaningful structures—such as emergent phases, critical points, or signatures of topological order—directly from quantum states, observables, or Hamiltonian matrices.

1. Unsupervised Learning Algorithms in Quantum Phase Detection

Unsupervised quantum phase detection leverages a variety of nonlinear, clustering, and dimensionality reduction algorithms that operate on representations of quantum states or measurement outcomes:

• Density-Based Spatial Clustering of Applications with Noise (DBSCAN) groups data points (eigenstates) according to density connectiveness set by a neighborhood radius (EPS) and a minimum number of neighbors (MinPoints). In application to quantum lattice models, DBSCAN distinguishes localized states (high-density clusters) from extended states (low-density, uniform spread), with critical states often occupying intermediate density regimes (Zheng et al., 19 Oct 2024).
• OPTICS extends DBSCAN by ordering points based on local “density reachability”, introducing the concepts of core distance (distance to MinPoints-th neighbor) and reachability distance. OPTICS more resolutely separates subtle or critical phases that do not fit crisp density thresholds, enhancing phase distinction along continuous transitions (Zheng et al., 19 Oct 2024).

Algorithm	Key Parameters	Phase Types Effectively Separated
DBSCAN	EPS, MinPoints	Extended vs. localized states
OPTICS	EPS, MinPoints	Extended, localized, critical states

Both algorithms are executed on the vector representations of eigenstates spanning the parameter space (e.g., disorder strength V in the Aubry-André–Harper or quasiperiodic p-wave model), using the Euclidean distance: $d = \sqrt{ \sum_{i=1}^n (x_{1,i} - x_{2,i})^2 }$ Phase diagrams are constructed by assigning cluster labels to states as a function of Hamiltonian parameters.

2. Phase Diagram Extraction in Quasiperiodic Lattices

In one-dimensional quasiperiodic systems, the quantum phase diagram is extracted by simulating eigenstates for a range of disorder strengths (V) and applying unsupervised clustering to the resulting wavefunctions:

• Aubry-André–Harper Model (AAH) uses the Hamiltonian:

$$\hat{H} = -\sum_{i=1}^{L-1} t (\hat{c}_i^\dagger \hat{c}_{i+1} + h.c.) + \sum_{i=1}^L V \cos(2\pi \alpha i + \phi) \hat{n}_i$$

with $\alpha = (\sqrt{5}-1)/2$, $L = 500$.

• Phase Assignments: Extended states exhibit inverse participation ratio (IPR) scaling as $1/L$, localized states have finite IPR, and critical states show power-law decay of the IPR slower than $1/L$.
• Visualization: Clustering outputs are color-coded over the (V, eigenstate index) phase diagram. Clear demarcations observed in clustering outputs (e.g., for DBSCAN and OPTICS) mirror boundaries obtained from direct calculation of IPR.

These unsupervised methods efficiently separate clusters corresponding to extended, localized, and critical states. Such approaches are also applicable to the quasiperiodic p-wave model (simulations with $L = 1000$).

3. Benchmarking Against Traditional Numerical Approaches

Traditional identification of quantum phases in quasiperiodic systems relies on the calculation of observables such as IPR or Lyapunov exponent ($\gamma$) for each eigenstate. Unsupervised learning results are quantitatively benchmarked against these traditional methods:

• IPR-based diagnostics precisely identify the V threshold for localization transitions (e.g., V=2t for the AAH model).
• Similarity Metric: The difference hash algorithm quantitatively compares machine learning phase diagrams to IPR-based diagrams.
  • DBSCAN achieves up to 98.4% similarity for the AAH model and over 95% for the p-wave model.
  • OPTICS produces slightly lower, but still significant, similarity (e.g. 90.6% for the AAH model, 62.5% for the p-wave model).
• Implication: DBSCAN exhibits particularly strong fidelity to conventional numerical diagonalization, validating its capacity to autonomously recover phase boundaries absent prior labels.

4. Applications and Broader Implications

The findings support broad applicability of unsupervised learning in scenarios where phase boundaries are complex, including many-body or interacting systems:

• Extension to Many-Body Systems: These methods can handle high-dimensional wavefunctions and the associated complexity of interacting phases.
• Noise and Disorder: The methods’ data-driven nature makes them robust to noisy data, suggesting utility for analysis of experimental results where disorder or measurement noise precludes traditional analytic techniques.
• Cluster Sensitivity: OPTICS, in particular, enhances detection of “critical” or intermediate phases, which are often missed by methods based solely on global observables like Lyapunov exponent.

Potential extensions include:

• Deployment with experimental post-processing for real quantum materials or cold atom experiments;
• Generalization to higher-dimensional or more complex Hamiltonians;
• Integration with further physical observables or hybrid architectures for enhanced interpretability.

5. Technical Considerations and Performance

Computational performance, tuning, and the effectiveness of the unsupervised clustering are governed by specific simulation and algorithmic choices:

• System Sizes: Eigenstates for the AAH model ($L=500$) and p-wave model ($L=1000$) are directly clustered.
• Parameter Tuning: DBSCAN is typically effective with EPS = 0.2, MinPoints = 1 for the AAH model; OPTICS succeeds with EPS = 0.2, MinPoints = 2.
• Visualization and Quantitative Metrics:
  • Figures illustrate both the clustering process and the comparative phase diagrams.
  • Table 1 aggregates similarity values for different algorithms and system models.
  • The most robust performance is found for DBSCAN, consistent across model architectures.

Model	Algorithm	Similarity (%)
AAH	DBSCAN	98.4
AAH	OPTICS	90.6
Quasiperiodic p-wave	DBSCAN	95.3
Quasiperiodic p-wave	OPTICS	62.5

6. Significance for Quantum Phase Detection

The demonstrated efficacy and quantitative agreement of unsupervised clustering methods with established numerical diagnostics validate their role as reliable, scalable tools for phase classification. Their data-driven design—

• Requires no prior physical or model-specific knowledge,
• Handles high-dimensional, raw wavefunction or eigenstate data,
• Produces phase diagrams closely matching traditional observables,
• Is readily applicable to experimental and noisy datasets,

—positions these methods for routine use in computational and experimental condensed matter research. The advances summarized in (Zheng et al., 19 Oct 2024) illustrate that unsupervised approaches are now competitive with—if not surpassing—many traditional analytic and numerical tools for the autonomous, interpretable classification of quantum phases in complex and disordered systems.