Abstract We extend the Prometheus framework from classical to quantum phase transitions, demonstrat- ing unsupervised discovery of quantum critical phenomena in the disordered transverse field Ising model (DTFIM). Building upon our previous work on classical spin systems [1, 2], we develop a quantum-aware variational autoencoder (Q-VAE) architecture that operates directly on quantum ground state wavefunctions obtained via exact diagonalization. The Q-VAE learns latent represen- tations that capture the essential physics of the quantum phase transition between paramagnetic and ferromagnetic phases, achieving detection of the quantum critical point hc = 1.00 ± 0.02 in the clean limit, consistent with the theoretical value hc/J = 1. We introduce disorder through random transverse fields h i ∼ Uniform[h−W, h+W ] and investigate the disorder-driven infinite-randomness fixed point characteristic of the random TFIM. Our framework successfully identifies the shift in critical behavior with increasing disorder strength and extracts effective critical exponents through finite-size scaling analysis. We formulate and test the hypothesis that the Q-VAE latent space structure reflects the activated dynamical scaling ln ξ ∼ |h − hc| −ψ with ψ ≈ 0.5 expected at the infinite-randomness fixed point. The Prometheus quantum discovery pipeline provides a systematic approach to exploring quantum phase diagrams in systems where analytical solutions are unavail- able, with potential applications to quantum materials, cold atom systems, and quantum computing platforms.
Abstract We extend the Prometheus framework from classical to quantum phase transitions, demonstrat- ing unsupervised discovery of quantum critical phenomena in the disordered transverse field Ising model (DTFIM). Building upon our previous work on classical spin systems [1, 2], we develop a quantum-aware variational autoencoder (Q-VAE) architecture that operates directly on quantum ground state wavefunctions obtained via exact diagonalization. The Q-VAE learns latent represen- tations that capture the essential physics of the quantum phase transition between paramagnetic and ferromagnetic phases, achieving detection of the quantum critical point hc = 1.00 ± 0.02 in the clean limit, consistent with the theoretical value hc/J = 1. We introduce disorder through random transverse fields h i ∼ Uniform[h−W, h+W ] and investigate the disorder-driven infinite-randomness fixed point characteristic of the random TFIM. Our framework successfully identifies the shift in critical behavior with increasing disorder strength and extracts effective critical exponents through finite-size scaling analysis. We formulate and test the hypothesis that the Q-VAE latent space structure reflects the activated dynamical scaling ln ξ ∼ |h − hc| −ψ with ψ ≈ 0.5 expected at the infinite-randomness fixed point. The Prometheus quantum discovery pipeline provides a systematic approach to exploring quantum phase diagrams in systems where analytical solutions are unavail- able, with potential applications to quantum materials, cold atom systems, and quantum computing platforms.
Abstract We present Prometheus, an unsupervised machine learning framework for discovering phase transitions and extracting critical exponents in three-dimensional spin systems. Building upon our previous work on the two-dimensional Ising model, we extend the variational autoencoder (VAE) approach to three-dimensional systems, demonstrating that the learned latent representations capture the essential physics of the ferromagnetic-paramagnetic phase transition. Our 3D convolutional VAE architecture achieves ≥70% accuracy in critical exponent extraction for the 3D Ising model, with the critical temperature $\Tc = 4.511 \pm 0.005$ in excellent agreement with the theoretical value $\Tc/J = 4.5115$. The framework incorporates a novel quantum discovery mechanism that enables systematic exploration of parameter spaces and automated detection of novel phase transition phenomena. We validate our approach through extensive finite-size scaling analysis, bootstrap confidence intervals, and comparison with established Monte Carlo results. The Prometheus framework provides a powerful tool for unsupervised exploration of phase diagrams in complex many-body systems where analytical solutions are unavailable.
We extend the Prometheus framework for unsupervised phase transition discovery from two-dimensional classical systems to three-dimensional classical systems and quantum many-body systems. Building upon preliminary observations from a 2D Ising model student abstract [Yee et al., 2026], we address two fundamental questions: (1) Does the framework scale to higher dimensions where exact solutions are unavailable? (2) Can it generalize to quantum phase transitions driven by quantum fluctuations rather than thermal fluctuations? For the 3D Ising model on lattices up to L=32, we achieve critical temperature detection within 0.01\% of literature values ($\Tc/J = 4.511 \pm 0.005$) and extract critical exponents with ≥70% accuracy, with statistical analysis correctly identifying the 3D Ising universality class (p=0.72). For quantum systems, we develop quantum-aware VAE (Q-VAE) architectures operating on complex-valued wavefunctions with fidelity-based loss functions, achieving 2\% accuracy in quantum critical point detection for the transverse field Ising model. For the disordered TFIM, we perform a consistency check of activated dynamical scaling $\ln ξ\sim |h - \hc|^{-ψ}$, extracting tunneling exponent ψ=0.48±0.08 consistent with theoretical predictions (ψ=0.5, Δχ2=12.3, p<0.001). This demonstrates that unsupervised learning can identify qualitatively different \emph{types} of critical behavior, serving as a consistency check on known IRFP physics.
Unsupervised Discovery of Intermediate Phase Order in the Frustrated J1-J2 Heisenberg Model via Prometheus Framework
The spin-1/2J1-J2 Heisenberg model on the square lattice exhibits a debated intermediate phase between Néel antiferromagnetic and stripe ordered regimes, with competing theories proposing plaquette valence bond, nematic, and quantum spin liquid ground states. We apply the Prometheus variational autoencoder framework -- previously applied to classical (2D, 3D Ising) and quantum (disordered transverse field Ising) phase transitions -- to systematically explore the J1-J2 phase diagram using a multi-scale approach. For L=4, we employ exact diagonalization with full wavefunction analysis via quantum-aware VAE. For larger systems (L=6,8), we introduce a reduced density matrix (RDM) based methodology using DMRG ground states, enabling scaling beyond the exponential barrier of full Hilbert space representation. Through dense parameter scans of J2/J1∈[0,1] and comprehensive latent space analysis, we identify the structure factor S(π,π) and S(π,0) as the dominant order parameters discovered by the VAE, with correlations exceeding ∣r∣>0.97. The RDM-VAE approach successfully captures the Néel-to-stripe crossover near J2/J1≈0.5--0.6, demonstrating that local quantum correlations encoded in reduced density matrices contain sufficient information for unsupervised phase discovery. This work establishes a scalable pathway for applying machine learning to frustrated quantum systems where full wavefunction access is computationally prohibitive.
Abstract We present an unsupervised machine learning framework for discovering phase transitions and order parameters in the two-dimensional Ising model without prior knowledge of the underlying physics. Using convolutional beta-variational autoencoders (β-VAE), we learn compressed representations that naturally separate ordered and disordered phases. Our physics-informed training procedures incorporate data augmentation respecting Ising symmetries, cosine annealing optimization, and progressive temperature curriculum learning. Experimental validation on Monte Carlo simulated configurations demonstrates automatic order parameter discovery with correlation ρ = 0.85 ± 0.04 to theoretical magnetization (95% CI [0.82, 0.88]) and critical temperature detection within 0.27% of the Onsager solution (p < 0.001). The method achieves 89% improvement over principal component analysis and 124% improvement over t-distributed stochastic neighbor embedding. Latent representations exhibit clear phase separation with physics consistency score S = 0.88 ± 0.03. Ablation studies reveal optimal performance at β = 1.0, balancing reconstruction fidelity with disentanglement. This framework enables automated identification of emergent phenomena in complex many-body systems, advancing machine learning applications in condensed matter physics.
What defines the operational boundary between quantum information and classical magnetism? We present a model-independent methodology for detecting characteristic scales in quantum systems through generalized susceptibility analysis χ(σ)=|d〈O〉/dσ|, where σ represents observation scales across spatial, temporal, and decoherence dimensions. Through comprehensive experiments on a Rigetti Ankaa-3 quantum processor, we identify an operational threshold at γc=0.6737±0.036 under tested noise profiles, exhibiting exceptional peak clarity κ=8.58—the sharpest signal observed across all phenomena. This pronounced transition, validated through bootstrap analysis and multiple susceptibility metrics, suggests critical-like behavior in the quantum-to-classical crossover, though we carefully refrain from claiming a thermodynamic phase transition. The framework successfully extracts characteristic scales without prior theoretical knowledge: correlation length ξ=8.00 qubits, distinct ordering timescales for ferromagnetic (tc=0.36) versus antiferromagnetic (tc=0.91) systems, and the quantum critical field hc=1.821 in the transverse-field Ising model. Robustness analysis confirms stability across smoothing parameters, methods, and metric definitions. Beyond providing actionable guidance for noisy intermediate-scale quantum-era quantum computing, our findings suggest that magnetic correlations exhibit characteristic transitions at specific decoherence scales—an operational perspective connecting quantum information theory and condensed matter physics. While framing this as methodological advance rather than fundamental physics, the identification of critical-like information-to-physics transitions provides practical diagnostics for quantum device characterization and optimization.
Despite the fact that a complete theoretical description of critical phenomena in connection with phase transitions has been well established through the renormalization-group theory, the microscopic nature of the phase transitions remains to be understood in a satisfactory way. For example, how does the interaction between particle/spins drive the system across distinct phases as parameters vary, and how do these particle/spins respond to parameter variations during this transition process? In this work, we investigate these questions through the paradigmatic example of quantum phase transitions (QPT) in the one-dimensional transverse Ising model (TIM). We first introduce 2L collective structures, referred to as patterns, for the TIM with L ferromagnetically interacting spins, and then analyze the contributions of these patterns to the system's states, e.g., the ground state, the first excited state, and so on, from which the analog of the QPT process between the disordered phase in the weakly coupling regime and the ferromagnetic phase in the strongly coupling regime is clearly identified at the interaction strength J_{c}=1. We systematically explore this process for small lattice sizes of L=6,8,10,12, whose ground-state energies are identical to those obtained by directly numerical exact diagonalization. Increasing the system size up to L=128, the actual QPT point located at J_{c}=1 in the thermodynamical limit is gradually approached. Our results show that the pattern picture is not only able to provide a microscopic process of phase transitions, but also of practical interest in analyzing analogues of QPT in diverse quantum simulation platforms.