Finding Quantum Many-Body Ground States with Artificial Neural Network

Jiaxin Wu; Wenjuan Zhang

arxiv: 1906.11216 · v1 · pith:PNCPHFHDnew · submitted 2019-06-26 · ❄️ cond-mat.dis-nn · physics.comp-ph· quant-ph

Finding Quantum Many-Body Ground States with Artificial Neural Network

Jiaxin Wu , Wenjuan Zhang This is my paper

Pith reviewed 2026-05-25 14:39 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn physics.comp-phquant-ph

keywords quantum many-body systemsground statesartificial neural networksunsupervised machine learningIsing modelHeisenberg modelwavefunction optimization

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The pith

An artificial neural network locates ground states of quantum many-body systems without assuming their functional form.

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

The paper introduces an unsupervised machine learning algorithm that employs an artificial neural network to determine the ground state of quantum many-body systems. This approach avoids any presupposed form for the wave function, enabling an unbiased optimization with accuracy that can be controlled. The method is tested on the one-dimensional Ising and Heisenberg models, yielding results that align closely with those from exact diagonalization. A reader would care because traditional methods for solving these ground states often require specific assumptions or become intractable for complex systems, and this provides a flexible alternative.

Core claim

We propose a new unsupervised machine learning algorithm utilizing artificial neural networks to find the ground state of a general quantum many-body system. Without assuming the specific forms of the eigenvectors, this algorithm can find the eigenvectors in an unbiased way with well controlled accuracy. As examples, we apply this algorithm to 1D Ising and Heisenberg models, where the results match very well with exact diagonalization.

What carries the argument

The unsupervised optimization of an artificial neural network to represent and minimize the energy of the many-body wavefunction.

If this is right

The algorithm applies to general quantum many-body systems.
Accuracy can be controlled during the optimization.
Results for the 1D Ising model match exact diagonalization.
Results for the 1D Heisenberg model match exact diagonalization.

Where Pith is reading between the lines

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

The representation could be applied to models in two or three dimensions where exact methods scale poorly.
Changing the loss function might allow the same network to target excited states instead of the ground state.
The approach could be hybridized with existing variational methods to handle larger system sizes.

Load-bearing premise

An artificial neural network can faithfully represent the ground-state wavefunction of a general quantum many-body system without any assumed functional form for the eigenvectors.

What would settle it

Applying the algorithm to the 1D Ising model and finding that the computed ground-state energy differs from the exact value obtained by diagonalization by more than the controlled accuracy bound.

Figures

Figures reproduced from arXiv: 1906.11216 by Jiaxin Wu, Wenjuan Zhang.

**Figure 1.** Figure 1: FIG. 1. (a) The traditional NN structure contains an input layer, some hidden layers, and an output layer. (b) Minimize the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2. (a) The typical behavior of the gain function [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. The acceptance rate under different random initial [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Two-point correlators of the first site with the other sites in the ground states with different system sizes. With periodic [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Ground state energy comparison between ED and [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

read the original abstract

Solving ground states of quantum many-body systems has been a long-standing problem in condensed matter physics. Here, we propose a new unsupervised machine learning algorithm to find the ground state of a general quantum many-body system utilizing the benefits of artificial neural network. Without assuming the specific forms of the eigenvectors, this algorithm can find the eigenvectors in an unbiased way with well controlled accuracy. As examples, we apply this algorithm to 1D Ising and Heisenberg models, where the results match very well with exact diagonalization.

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 sketches an unsupervised ANN method for quantum ground states that matches exact diagonalization on two 1D models but supplies almost no supporting details or tests beyond that.

read the letter

The core claim is that an artificial neural network can locate ground states of general quantum many-body systems in an unsupervised way without built-in assumptions about the wavefunction form, and that accuracy stays well controlled. They show this on the 1D Ising and Heisenberg chains where the numbers line up with exact diagonalization. That match on the simple cases is the only concrete evidence offered. The approach is presented as new, though the abstract alone does not clarify how it differs from earlier variational neural-network work on quantum states. What stands out is the attempt to keep the method unbiased and general. The main weakness is that nothing is shown about the actual network, the loss function, the optimization steps, error bars, or scaling with system size. The examples stay in one dimension with modest entanglement, so the central assertion that the method works for general systems rests on an untested assumption about representation power. No discussion appears of sign problems, higher dimensions, or failure modes. Readers already working on machine-learning approaches to many-body physics might find the high-level idea worth a quick look, but the lack of methods and broader tests makes it hard to judge whether the result adds anything reproducible. I would not bring this to a reading group in its current form and would not cite it. It does not look ready for peer review; the claims need the missing technical backbone and more demanding benchmarks before a referee should spend time on it.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes an unsupervised machine learning algorithm that employs artificial neural networks to locate the ground states of general quantum many-body systems. It asserts that the method determines eigenvectors in an unbiased manner with controlled accuracy, without presupposing specific functional forms for the wavefunctions. The approach is illustrated on the one-dimensional Ising and Heisenberg models, where the obtained results are stated to agree well with exact diagonalization.

Significance. If the algorithm can be shown to deliver unbiased representations with quantifiable accuracy beyond the simplest cases, it would constitute a useful addition to the toolkit for quantum many-body problems by exploiting the flexibility of neural-network ansatzes. The unsupervised character and absence of assumed forms are positive features. At present, however, the evidential base is confined to one-dimensional models, limiting the immediate impact.

major comments (2)

[Abstract] Abstract: The central claim that the algorithm finds eigenvectors 'in an unbiased way with well controlled accuracy' for a 'general quantum many-body system' is not supported by the reported evidence, which is restricted to 1D Ising and Heisenberg chains; no tests on higher-dimensional lattices, frustrated systems, or models with nontrivial sign structure are provided to substantiate generality.
[Abstract] The manuscript supplies no derivation of the training procedure, loss function, or error-control mechanism that would establish how accuracy is 'well controlled' independently of the specific model; the agreement with exact diagonalization is asserted but lacks quantitative metrics, convergence data, or scaling analysis with system size.

minor comments (1)

Notation for the neural-network parameters and the precise form of the variational wavefunction should be introduced explicitly before the numerical examples.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below and outline revisions that will be incorporated to improve the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim that the algorithm finds eigenvectors 'in an unbiased way with well controlled accuracy' for a 'general quantum many-body system' is not supported by the reported evidence, which is restricted to 1D Ising and Heisenberg chains; no tests on higher-dimensional lattices, frustrated systems, or models with nontrivial sign structure are provided to substantiate generality.

Authors: We agree that the numerical evidence is confined to 1D models and that the phrasing 'general quantum many-body system' overstates the current validation. The algorithm is formulated without model-specific assumptions, but demonstrations remain limited. We will revise the abstract and introduction to qualify the scope, explicitly state that validation is on 1D chains, and add a paragraph on planned extensions to higher dimensions and sign-problem cases. revision: yes
Referee: [Abstract] The manuscript supplies no derivation of the training procedure, loss function, or error-control mechanism that would establish how accuracy is 'well controlled' independently of the specific model; the agreement with exact diagonalization is asserted but lacks quantitative metrics, convergence data, or scaling analysis with system size.

Authors: The main text contains a derivation of the unsupervised loss (variational energy expectation value) and the stochastic gradient updates used for training. However, we accept that quantitative support for controlled accuracy is insufficient. We will add explicit convergence plots versus training steps, tables of relative energy errors versus exact diagonalization, and finite-size scaling of the obtained ground-state energy to demonstrate error control. revision: yes

Circularity Check

0 steps flagged

No circularity: algorithm proposed and externally validated on 1D models

full rationale

The paper proposes an unsupervised ANN-based algorithm to locate ground states without assuming eigenvector forms, then validates it by direct numerical comparison to exact diagonalization on the 1D Ising and Heisenberg chains. No derivation step reduces by construction to its own inputs, no fitted parameter is relabeled as a prediction, and no load-bearing premise rests on a self-citation chain. The central claim is an empirical demonstration of matching accuracy rather than a tautological equivalence, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5605 in / 868 out tokens · 27456 ms · 2026-05-25T14:39:30.196777+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 2 internal anchors

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Finding Quantum Many-Body Ground States with Artificial Neural Network

proposed a method using convolutional neural net- works. In condensed matter physics, one of the main hurdles is to ﬁnd the eigenstates of interacting systems with rea- sonable system sizes. In the presence of interactions, the dimension of the Hilbert space grows exponentially with the system size, which prevents us from solving the Hamiltonian exactly e...

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Y. Wu, P. Zhang, H. Shen, and H. Zhai, Visualizing a neural network that develops quantum perturbation the- ory, Physical Review A 98, 010701 (2018)

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S. J. Wetzel, Unsupervised learning of phase transitions: From principal component analysis to variational autoen- coders, Physical Review E 96, 022140 (2017)

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Quantum phase recognition via unsupervised machine learning

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Finding Quantum Many-Body Ground States with Artificial Neural Network

proposed a method using convolutional neural net- works. In condensed matter physics, one of the main hurdles is to ﬁnd the eigenstates of interacting systems with rea- sonable system sizes. In the presence of interactions, the dimension of the Hilbert space grows exponentially with the system size, which prevents us from solving the Hamiltonian exactly e...

work page internal anchor Pith review Pith/arXiv arXiv 1906

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Torlaiand R

G. Torlaiand R. G. Melko, Learning thermodynamics with boltzmann machines, Physical Review B 94, 165134 (2016)

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Y. Wu, P. Zhang, H. Shen, and H. Zhai, Visualizing a neural network that develops quantum perturbation the- ory, Physical Review A 98, 010701 (2018)

work page 2018

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S. J. Wetzel, Unsupervised learning of phase transitions: From principal component analysis to variational autoen- coders, Physical Review E 96, 022140 (2017)

work page 2017

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Quantum phase recognition via unsupervised machine learning

P. Broecker, F. F. Assaad, and S. Trebst, Quantum phase recognition via unsupervised machine learning, arXiv preprint arXiv:1707.00663 (2017)

work page internal anchor Pith review Pith/arXiv arXiv 2017

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G. Carleoand M. Troyer, Solving the quantum many- body problem with artiﬁcial neural networks, Science 355, 602 (2017)

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