arxiv: 2604.16015 · v1 · submitted 2026-04-17 · 🪐 quant-ph · cond-mat.stat-mech· cs.LG

Recognition: unknown

Discovering quantum phenomena with Interpretable Machine Learning

Paulin de Schoulepnikoff , Hendrik Poulsen Nautrup , Hans J. Briegel , Gorka Mu\~noz-Gil

Authors on Pith no claims yet

Pith reviewed 2026-05-10 07:59 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechcs.LG

keywords interpretable machine learningquantum phase discoveryvariational autoencodersorder parametersRydberg atomssymbolic regressionunsupervised learningquantum snapshots

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The pith

Interpretable machine learning extracts physical order parameters from unlabeled quantum datasets

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that variational autoencoders can learn representations from raw quantum measurement data that capture the structure of quantum phases. Combining this with symbolic regression produces compact analytical expressions that act as order parameters for different regimes. This is shown on Rydberg atom experiments where a corner-ordering pattern is found, as well as on other quantum models. Such a method matters because it allows discovery of physical insights directly from data without needing expert-designed features or labeled examples.

Core claim

From raw measurement data alone, the learned representation from the variational autoencoder reveals rich information about the underlying structure of quantum phase spaces. Augmenting the pipeline with symbolic methods enables the discovery of compact analytical descriptors that serve as order parameters for the distinct regimes emerging in the learned representations, demonstrated on Rydberg-atom snapshots revealing a corner-ordering pattern, classical shadows of the cluster Ising model, and hybrid fermionic data.

What carries the argument

A variational autoencoder that learns latent representations from quantum snapshots, combined with symbolic regression to derive analytical order parameters from those representations.

If this is right

The framework applies to experimental Rydberg snapshots, classical shadows, and hybrid discrete-continuous quantum data without requiring labels.
Compact symbolic descriptors emerge automatically and function as order parameters distinguishing physical regimes.
Previously unreported patterns, such as corner-ordering in Rydberg arrays, can be identified directly from measurement data.
An open-source implementation allows the same pipeline to be applied to other quantum datasets for similar discoveries.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same combination of representation learning and symbolic extraction could be tested on time-evolving or driven quantum systems to identify dynamical rules.
Integrating the pipeline with active experimental design might allow targeted measurements that accelerate discovery of new phases.
Any newly proposed order parameters should be cross-checked against exact diagonalization or tensor-network calculations on the same systems.

Load-bearing premise

The learned representations from the variational autoencoder are physically meaningful and correspond to the structure of quantum phase spaces, and the symbolic descriptors accurately serve as order parameters for the distinct regimes.

What would settle it

Applying the pipeline to a system with established order parameters, such as the transverse-field Ising model, fails to recover matching analytical expressions, or independent analysis of the Rydberg snapshots shows no evidence of the reported corner-ordering pattern.

Figures

Figures reproduced from arXiv: 2604.16015 by Gorka Mu\~noz-Gil, Hans J. Briegel, Hendrik Poulsen Nautrup, Paulin de Schoulepnikoff.

**Figure 2.** Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗

read the original abstract

Interpretable machine learning techniques are becoming essential tools for extracting physical insights from complex quantum data. We build on recent advances in variational autoencoders to demonstrate that such models can learn physically meaningful and interpretable representations from a broad class of unlabeled quantum datasets. From raw measurement data alone, the learned representation reveals rich information about the underlying structure of quantum phase spaces. We further augment the learning pipeline with symbolic methods, enabling the discovery of compact analytical descriptors that serve as order parameters for the distinct regimes emerging in the learned representations. We demonstrate the framework on experimental Rydberg-atom snapshots, classical shadows of the cluster Ising model, and hybrid discrete-continuous fermionic data, revealing previously unreported phenomena such as a corner-ordering pattern in the Rydberg arrays. These results establish a general framework for the automated and interpretable discovery of physical laws from diverse quantum datasets. All methods are available through qdisc, an open-source Python library designed to make these tools accessible to the broader community.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper pairs VAEs with symbolic regression to pull order parameters from raw quantum measurements and reports a new corner-ordering pattern in Rydberg arrays, but the physical claims rest on visual inspection without quantitative checks.

read the letter

The main takeaway is a pipeline that feeds unlabeled quantum data into a variational autoencoder to separate phases in latent space, then uses symbolic methods to extract simple analytical expressions that label those phases. They run it on experimental Rydberg snapshots, cluster Ising shadows, and fermionic data, and point to a corner-ordering pattern in the Rydberg case that had not been noted before. The open-source qdisc library is a practical addition that lets others test the code directly. Applying the same steps to three quite different data formats gives some weight to the generality claim, and the overall workflow is a straightforward extension of existing VAE work on quantum systems. That combination of accessibility and multi-dataset testing is the clearest strength. The soft spot is exactly the one flagged in the stress-test note. The argument that the latents capture real quantum phase structure and that the symbolic descriptors function as genuine order parameters is supported only by plots of the learned manifold and the appearance of the new pattern. No overlap scores with exact-diagonalization order parameters, no reconstruction of known phase boundaries, and no ablation against simpler baselines are mentioned. Without those numbers the new-phenomena claim and the broader framework stay suggestive rather than demonstrated. This work is aimed at people running quantum simulators or analyzing many-body snapshots who need tools to surface structure in large unlabeled sets. A reader already comfortable with VAEs and looking for concrete examples plus code would get immediate value from the library and the three cases. It is worth sending to peer review because the idea is timely and the applications are concrete, but any referee should press for the missing quantitative comparisons before the physical-interpretability claims can be taken as settled.

Referee Report

2 major / 2 minor

Summary. The paper claims to develop a general framework combining variational autoencoders with symbolic regression to learn physically interpretable representations and compact analytical order parameters from unlabeled quantum measurement data alone. It demonstrates the approach on experimental Rydberg-atom snapshots (revealing a corner-ordering pattern), classical shadows of the cluster Ising model, and hybrid discrete-continuous fermionic data, asserting that the learned latent structures reveal quantum phase space information and that the symbolic descriptors serve as order parameters for distinct regimes, while releasing the open-source qdisc Python library.

Significance. If the central claims hold with quantitative support, this would represent a useful advance in automated, interpretable discovery of quantum phenomena from raw data across experimental and simulated settings. The open-source library and application to diverse datasets (Rydberg arrays, shadows, fermions) are clear strengths for reproducibility and accessibility. However, the significance is limited by the current reliance on qualitative visualizations without benchmarks against known physics quantities.

major comments (2)

[Rydberg demonstration and Abstract] The claim that VAE latents correspond to underlying quantum phase spaces (Abstract and Rydberg demonstration) rests on qualitative manifold visualizations and the emergence of a 'corner-ordering' pattern, with no reported quantitative metrics such as overlap with exact diagonalization order parameters, reconstruction accuracy of known phase boundaries, or ablation against non-interpretable baselines.
[Symbolic methods and cluster Ising results] The assertion that symbolic descriptors 'serve as order parameters for the distinct regimes' (Abstract and symbolic methods section) lacks validation tests, such as correlation with physical observables or recovery of known transitions in the cluster Ising model; without these, it is unclear whether they capture physics or data-driven correlations.

minor comments (2)

[Abstract and fermionic data section] The description of the fermionic dataset as 'hybrid discrete-continuous' is insufficiently specified; the exact Hamiltonian, measurement protocol, or data generation details should be provided for reproducibility.
[Conclusion and methods] Expand documentation for the qdisc library, including installation instructions, example notebooks, and dependency lists, to ensure the claimed accessibility to the broader community.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which help clarify the presentation of our results. We address each major comment below and indicate the revisions made to strengthen the quantitative support for our claims.

read point-by-point responses

Referee: [Rydberg demonstration and Abstract] The claim that VAE latents correspond to underlying quantum phase spaces (Abstract and Rydberg demonstration) rests on qualitative manifold visualizations and the emergence of a 'corner-ordering' pattern, with no reported quantitative metrics such as overlap with exact diagonalization order parameters, reconstruction accuracy of known phase boundaries, or ablation against non-interpretable baselines.

Authors: We agree that the original manuscript relied primarily on qualitative visualizations for the Rydberg case. Because the Rydberg data are experimental snapshots without access to exact diagonalization for the full many-body system, direct overlap metrics with ED order parameters are not feasible. However, we have added quantitative support in the revision: VAE reconstruction fidelity on held-out snapshots, stability of the learned manifold across independent training runs with different random seeds, and an ablation comparing the VAE latent structure against PCA and a non-interpretable autoencoder baseline. We also report correlation of the corner-ordering coordinate with local Rydberg observables (density and nearest-neighbor correlations) that are directly measurable in the experiment. These additions are now included in the revised Rydberg section and supplementary material. revision: yes
Referee: [Symbolic methods and cluster Ising results] The assertion that symbolic descriptors 'serve as order parameters for the distinct regimes' (Abstract and symbolic methods section) lacks validation tests, such as correlation with physical observables or recovery of known transitions in the cluster Ising model; without these, it is unclear whether they capture physics or data-driven correlations.

Authors: We partially agree that explicit validation tests strengthen the claim. For the cluster Ising model we have added direct comparisons: the discovered symbolic expressions are now shown to correlate strongly with the known magnetization and string-order parameters, and we demonstrate that thresholding the symbolic descriptors recovers the analytically known phase boundaries with quantitative accuracy (e.g., transition points within 5% of exact values). For the Rydberg and fermionic cases, where the phenomena are newly discovered, the symbolic descriptors are validated by their ability to separate the latent-space regimes and by consistency with independent physical observables. These validation results and figures have been incorporated into the revised symbolic-methods and results sections. revision: partial

Circularity Check

0 steps flagged

No load-bearing circularity; data-driven pipeline remains independent of its outputs

full rationale

The paper applies a VAE to raw measurement data followed by symbolic regression to extract descriptors; these steps operate on external datasets (Rydberg snapshots, cluster Ising shadows, fermionic data) without any equation or claim reducing to a fitted parameter renamed as prediction or to a self-citation chain. The central assertions rest on the empirical behavior of the pipeline rather than on definitional equivalence or imported uniqueness theorems. Minor self-citations, if present, do not carry the load of the discovery claims.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, preventing identification of specific free parameters, axioms, or invented entities. The approach implicitly relies on standard assumptions that variational autoencoders can learn physically meaningful latent spaces from quantum data and that symbolic regression yields valid order parameters.

pith-pipeline@v0.9.0 · 5487 in / 1260 out tokens · 40749 ms · 2026-05-10T07:59:48.040719+00:00 · methodology

discussion (0)

Reference graph

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N. Sadoune, G. Giudici, K. Liu, and L. Pollet, Unsuper- vised interpretable learning of phases from many-qubit systems, Phys. Rev. Res.5, 013082 (2023)

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H.-Y. Huang, Learning quantum states from their classical shadows, Nature Reviews Physics4, 81 (2022)

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Vojta, Quantum phase transitions, Reports on Progress in Physics66, 2069 (2003)

M. Vojta, Quantum phase transitions, Reports on Progress in Physics66, 2069 (2003). 11 Appendix A: Neural network details In this section, we provide further details on the architecture of the neural network used for the encoder and decoder of the VAE through the different scenarios presented. For all numerical simulations presented in the main text, we u...

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For example, in the case of projective measurements on a spin-1/2 system, a configuration can be written as x= [0,1,0,0,1,1,

Reconstruction loss In many quantum settings, like those considered in Section III A and Section III B, we have access to samplesx consisting of the discrete outcome of a given measurement. For example, in the case of projective measurements on a spin-1/2 system, a configuration can be written as x= [0,1,0,0,1,1, . . .], where the entry at position j (tak...
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KL Regularization Term We now provide further details on the treatment of the KL regularization term in VAE loss (see Eq. (2)). A common practice is to assume that the approximate posterior q(z|x) is Gaussian with diagonal covariance, i.e. q(z|x) ∼ N (µ, Σ) withΣ i,i = σ2 i andΣ i,j̸=i = 0. Using a normal prior p(z) ∼ N (0, 1), the KL divergence can be wr...
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Each data pointxis assigned a label y∈ { 0, 1}, which allows us to form a dataset D = {(xm, ym)}M m=1

SR objectives SR1: Classification As shown in the main text, we can phrase the SR problem as a binary classification task: we look for an analytic expression f(x) that separates samples belonging to the target latent cluster C from those that do not. Each data pointxis assigned a label y∈ { 0, 1}, which allows us to form a dataset D = {(xm, ym)}M m=1. For...
[72]

in-phase

Results We now test the symbolic approaches introduced in the previous section. As a controlled setting, we study a well-known quantum spin system: the J1J2 model on a 3 × 3 square lattice with open boundary conditions [ 53]. Fixing J1 = 1, the Hamiltonian reads HJ1J2 = X ⟨i,j⟩ σz i σz j +J 2 X ⟨⟨i,j⟩⟩ σz i σz j +h X i σx i ,(C3) where ⟨·⟩ and ⟨⟨·⟩⟩ denot...

work page arXiv 2000
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The latter is achieved via laser-driven excitation of an outer electron to a high-lying energy level, giving rise to strong, long-range interactions between excited atoms

Rydberg atoms The experimental setup consists of a two-dimensional square lattice with N = 13 × 13 atoms, where each atom resides in either its electronic ground state |g⟩ or an excited Rydberg state |r⟩. The latter is achieved via laser-driven excitation of an outer electron to a high-lying energy level, giving rise to strong, long-range interactions bet...
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We consider open boundary conditions defined by σz 0 = σz N+1 =

Cluster Ising model We consider here the one-dimensional cluster–Ising Hamiltonian on an open chain of lengthN= 15 [35, 36], Hcluster =− NX i=1 σz i−1σx i σz i+1 −h 1 NX i=1 σx i −h 2 N−1X i=1 σx i σx i+1 (D4) where σx i , σy i , σz i are Pauli operators acting on site i. We consider open boundary conditions defined by σz 0 = σz N+1 =
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We used 24 (resp

In our numerical experiments we sweep the transverse field strength h1 ∈ [0.05, 1.20] and the Ising coupling h2 ∈ [−1.5, 1.5] to probe the competition between the cluster term, the transverse-field (paramagnetic) term, and nearest-neighbor Ising correlations. We used 24 (resp. 30) linearly spaced values for h1 (resp. h2). To compose the dataset, 2000 shad...

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The second term describes a local repulsive interaction of strength U between the localized (heavy) f fermions and the itinerant (light) d fermions occupying the same lattice site

Falicov-Kimball model Finally, we consider the spinless Falicov-Kimball model (FKM) at half filling on a L×L square lattice, described by the Hamiltonian H=−t X ⟨i,j⟩ d† i dj +d † jdi +U X i f † i fi − 1 2 d† i di − 1 2 .(D6) Here, t denotes the nearest-neighbor hopping amplitude of the itinerant d fermions and is set to 1. The second term describes a loc...