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arxiv: 2606.28911 · v1 · pith:QK4OVYSWnew · submitted 2026-06-27 · 💻 cs.LG · cond-mat.mtrl-sci· cs.DC· physics.comp-ph

MALOQ: Massively Accelerated Learning of Operators for Quantum Transport

Manasa Kaniselvan , Alexander Maeder , Denghui Lu , Alexandros Nikolaos Ziogas , Mathieu Luisier This is my paper

Pith reviewed 2026-06-30 09:59 UTC · model grok-4.3

classification 💻 cs.LG cond-mat.mtrl-scics.DCphysics.comp-ph
keywords machine learningquantum transportHamiltonian predictionequivariant neural networksdistributed trainingDFT matriceslarge-scale atomic systemselectronic structure
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The pith

MALOQ trains equivariant models on the largest Hamiltonian datasets to predict quantum operators for systems up to 100,000 atoms while cutting time-per-epoch by over 30 percent.

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

The paper presents MALOQ as a framework that trains machine-learned models to output density functional theory Hamiltonian and density matrices for atomic systems ranging from a few atoms to 100,000 atoms. It builds on an SO(2)-equivariant neural network backbone and adds custom kernels to process high-rank matrix data together with an edge-wise distribution scheme that spreads atomic graphs across many GPUs. The result is faster training on the biggest available molecular datasets and the ability to run inference on material graphs whose size is no longer fixed by the training procedure. Demonstrations on the Alps supercomputer show training and inference scaling to thousands of atoms across up to 256 GPUs.

Core claim

MALOQ is an application that trains SO(2)-equivariant models on large molecular Hamiltonian datasets to predict electronic-structure matrices for systems of a few to 100k atoms, using custom data-processing kernels for high-rank matrices and a scalable edge-wise distribution of atomic graphs, thereby reducing time-per-epoch by more than 30 percent relative to molecule-wise distribution and enabling inference on material graphs of arbitrary size.

What carries the argument

SO(2)-equivariant backbone architecture together with custom data-processing kernels for high-rank Hamiltonian matrices and an edge-wise distribution scheme for atomic graphs.

If this is right

  • Training on the largest available molecular Hamiltonian datasets becomes feasible with more than 30 percent lower time per epoch.
  • Inference runs on material graphs whose total atom count is no longer constrained by the size of the training examples.
  • Electronic-structure calculations extend to system sizes previously unreachable by direct DFT methods.
  • Scalable training and inference are demonstrated for 3,000-12,000 atom systems on up to 256 GPUs.

Where Pith is reading between the lines

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

  • The same distribution and kernel approach could be reused for other high-rank quantum operators such as density matrices or current operators.
  • Once trained, the model could serve as a fast surrogate inside device-scale transport simulations that mix atomistic and continuum regions.
  • The arbitrary-size inference property removes the need to retrain when moving from molecular test sets to periodic or nanostructured materials.

Load-bearing premise

The SO(2)-equivariant model with the custom kernels learns accurate high-rank Hamiltonian predictions across many atomic elements and system sizes up to 100k atoms without substantial accuracy loss.

What would settle it

A direct accuracy comparison on a held-out set of 50,000-atom structures containing underrepresented elements that shows prediction errors growing beyond the tolerance reported for smaller training systems.

Figures

Figures reproduced from arXiv: 2606.28911 by Alexander Maeder, Alexandros Nikolaos Ziogas, Denghui Lu, Manasa Kaniselvan, Mathieu Luisier.

Figure 1
Figure 1. Figure 1: Applications of electronic structure learning in molecules (top) and materials (bottom). The goal is to learn the mapping [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Custom kernels for data processing and matrix reconstruction (a) Schematic of the different steps along the inference process, highlighting (in blue boxes) the two non￾learnable model components and in black frame the location where the ‘Matrix→Label’ conversion occurs. (b) Illustration showing the mapping between the matrix and label represen￾tations of data for a toy H2O molecule with a minimal basis (Ox… view at source ↗
Figure 3
Figure 3. Figure 3: Fused Triton kernel for Wigner-D matrix con￾struction. (a) Schematic of the inference process highlighting the location where rotations occur in the MALOQ application (blue box with thick black frame). (b) These Wigner-D ma￾trices rotate the node/edge embedding ni/eij of every edge rˆij from the atomic graph to align it with the +y-axis. (c) Doing this requires first computing the Euler angles (α, β, γ) of… view at source ↗
Figure 5
Figure 5. Figure 5: Training runtime comparison under different dis￾tribution schemes. (a) Dataset used for training. It consists of 128 randomly-selected molecules from the OMol electrolytes dataset and their corresponding H. (b) Workload distribution across ranks quantified in # atoms and # edges, for three different distribution schemes (atom-wise, edge-wise, Metis), compared to the undistributed case in which 128 molecule… view at source ↗
Figure 4
Figure 4. Figure 4: Custom communication scheme for graph-level distribution. (a) Schematic of the inference process showing where the communication occurs within MALOQ, i.e., at the start of each node/edge block within a eSCN convolution layer (black frames). (b) Illustration of the communication pattern required to assemble the messages mi→j within a toy H2O molecule. (c) Illustration of the send/recv operation implementati… view at source ↗
Figure 6
Figure 6. Figure 6: Embedding communication and processing through￾put. (a) Measured intra-/inter-node communication throughput of node embeddings ni with embedding size E=128. (b) Processing throughput for eSCN convolutions as a function of the number of messages mi→j for different Lmax. Each data point corresponds to the median of 20 measurements. ‘binning’ molecules into partitions to balance the total number of atoms/edge… view at source ↗
Figure 7
Figure 7. Figure 7: Strong scaling of MALOQ over three irregular molecular datasets from Table II with different Lmax. (a) Forward pass. (b) Backward pass. (c) Total time per batch. A total of 128 molecules from each dataset is placed in each batch, which is distributed according to an edge-wise partition. The data plotted is the median over 200 epochs (the first 20 are discarded), and the shaded areas show variance over runt… view at source ↗
Figure 8
Figure 8. Figure 8: Scalable training and inference on large materials. (a) Strong scaling of MALOQ’s training on large material datasets. The times for the forward (orange) and backward (blue) passes are reported for three different partitioning of an amorphous 3000-atom HfO2 structure (shown in inset). (b) Weak scaling over inference for the same structure as in (a). The problem size is increased according to two tiling sch… view at source ↗
read the original abstract

Machine-learned (ML) operator models can be trained to predict density functional theory (DFT) Hamiltonian/density matrices at significantly reduced computational cost, thus extending electronic-structure calculations to previously unfeasible scales. Here, we introduce MALOQ (Massively Accelerated Learning of Operators for Quantum Transport), an application built to train on and predict electronic-structure matrices for systems made of few to 100k atoms, described by large basis sets, and covering a wide range of atomic elements. Based on a state-of-the-art, SO(2)-equivariant backbone architecture, MALOQ provides (i) custom data-processing kernels to handle high-rank Hamiltonian matrix data and (ii) a scalable edge-wise distribution of atomic graph(s). Trained on the largest molecular Hamiltonian datasets available today, it reduces time-per-epoch by over 30% compared to a molecule-wise-distributed framework, and enables inference on material graphs of arbitrary size. We demonstrate scalable training and inference for 3,000-12,000 atoms on the Alps supercomputer, up to 192 GPUs and 256 GPUs, respectively.

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.

Referee Report

2 major / 2 minor

Summary. The paper introduces MALOQ, a framework built on an SO(2)-equivariant backbone with custom data-processing kernels and edge-wise graph distribution for training ML models to predict DFT Hamiltonian/density matrices. It claims >30% reduction in time-per-epoch versus molecule-wise distribution, scalable training/inference on 3k–12k atom systems (up to 192/256 GPUs on Alps), and the ability to handle arbitrary-size material graphs up to 100k atoms across diverse elements and large basis sets.

Significance. If accuracy on large systems is validated, the work could enable electronic-structure calculations at previously inaccessible scales for quantum transport. The reported engineering contributions (custom kernels, edge-wise distribution) address real bottlenecks in distributed training on high-rank matrix data, and the scaling demonstrations on a production supercomputer are concrete.

major comments (2)
  1. [Abstract] Abstract: the central utility claim—that the model learns accurate high-rank Hamiltonian predictions for systems up to 100k atoms without substantial accuracy loss—is unsupported; no MAE on matrix elements, eigenvalue errors, transport quantities, or DFT ground-truth comparisons are reported for any system size, including the 3k–12k atom demonstrations.
  2. [Abstract] Abstract and scaling sections: the assertion of inference on material graphs of arbitrary size rests on extrapolation from 3k–12k atom results; no evidence (accuracy metrics, ablation on graph size, or tests on non-molecular periodic systems) is provided to substantiate that accuracy holds at 100k atoms or for arbitrary-size inference.
minor comments (2)
  1. The manuscript should include at least one table or figure reporting quantitative accuracy (e.g., MAE per matrix element or band-structure error) against DFT reference for the largest demonstrated systems.
  2. Clarify the precise definition of 'time-per-epoch' reduction (wall-clock, per-GPU, or communication overhead) and the baseline molecule-wise framework implementation details.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the referee's constructive feedback on our manuscript. We appreciate the identification of areas where the abstract and scaling claims require additional support or clarification. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central utility claim—that the model learns accurate high-rank Hamiltonian predictions for systems up to 100k atoms without substantial accuracy loss—is unsupported; no MAE on matrix elements, eigenvalue errors, transport quantities, or DFT ground-truth comparisons are reported for any system size, including the 3k–12k atom demonstrations.

    Authors: We agree that the abstract overstates the validated accuracy for large systems. The current work emphasizes the engineering contributions for scalable training and inference on high-rank matrix data, with accuracy inherited from the SO(2)-equivariant backbone trained on available datasets. To address this, the revised manuscript will include MAE on matrix elements, eigenvalue errors, and transport quantity comparisons for the 3k–12k atom demonstrations, along with explicit DFT ground-truth references where available. The 100k-atom claim will be qualified as prospective. revision: yes

  2. Referee: [Abstract] Abstract and scaling sections: the assertion of inference on material graphs of arbitrary size rests on extrapolation from 3k–12k atom results; no evidence (accuracy metrics, ablation on graph size, or tests on non-molecular periodic systems) is provided to substantiate that accuracy holds at 100k atoms or for arbitrary-size inference.

    Authors: We acknowledge that the arbitrary-size inference claim relies on the design of the edge-wise distribution, which removes molecule-wise batching constraints and permits processing of graphs of any size in principle. The 3k–12k atom results demonstrate linear scaling behavior up to the tested regime. In the revision, we will update the abstract and scaling sections to state that accuracy at 100k atoms and on non-molecular periodic systems is extrapolated rather than directly validated, and we will add a dedicated limitations paragraph on this point. revision: yes

Circularity Check

0 steps flagged

No circularity detected; claims rest on implementation and empirical scaling demos

full rationale

The paper describes an ML application (MALOQ) using an SO(2)-equivariant GNN backbone plus custom kernels for matrix data and edge-wise graph distribution. Performance claims (30% epoch time reduction, arbitrary-size inference) are presented as outcomes of these engineering choices, with concrete scaling results shown only for 3k-12k atom systems. No derivations, fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations appear in the provided text. The 100k-atom accuracy claim is an unverified extrapolation but does not reduce any result to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the contribution is framed as an engineering application of existing equivariant ML techniques to quantum transport data.

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