MALOQ: Massively Accelerated Learning of Operators for Quantum Transport

# MALOQ: Massively Accelerated Learning of Operators for Quantum Transport
Source: [https://arxiv.org/html/2606.28911](https://arxiv.org/html/2606.28911)
###### 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 introduceMALOQ\(Massively Accelerated Learning of Operators for Quantum Transport\), 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\.

## IIntroduction

Machine\-learned \(ML\) models are accelerating computational materials science research by extending the reach of atomistic simulations to previously infeasible scales\[[52](https://arxiv.org/html/2606.28911#bib.bib73)\]\. So far, these models have focused on learningpropertiesat the molecular and atomic level\. This includes the class of ML Interatomic Potentials \(MLIPs\), which predict the molecular energies and atomic forces used in molecular dynamics \(MD\) simulations\. State\-of\-the\-art, graph\-based ‘materials foundation models’ are now capable of generating these quantities for arbitrary molecules and complex material geometries at quasi\-DFT\-level accuracy\[[3](https://arxiv.org/html/2606.28911#bib.bib9),[46](https://arxiv.org/html/2606.28911#bib.bib10),[31](https://arxiv.org/html/2606.28911#bib.bib76)\]\.

An emerging class of ML materials models extends these architectures toward electronic\-level quantities, defined byoperators\[[14](https://arxiv.org/html/2606.28911#bib.bib79),[20](https://arxiv.org/html/2606.28911#bib.bib12),[33](https://arxiv.org/html/2606.28911#bib.bib71),[30](https://arxiv.org/html/2606.28911#bib.bib11)\]\. Rather than learning atom\-centered properties, these ‘operator models’ are directly trained on the matrices encoding interactions between the inter\- and intra\-atomic orbitals of a given system\. The most commonly targeted operators are the Hamiltonian/Fock \(𝐇\\mathbf\{H\}\) and density \(𝐏\\mathbf\{P\}\) matrices, which describe the spatial and energetic distribution of states that electrons can occupy within the atomic structure considered\. Recent works have extended these principles to learn other operators that can be represented in an atomic orbital basis, either directly, such as the Green’s function \(inverse of𝐇\\mathbf\{H\}\)\[[43](https://arxiv.org/html/2606.28911#bib.bib2)\], or indirectly, e\.g\., lower\-rank Hessian\[[8](https://arxiv.org/html/2606.28911#bib.bib14)\]matrices which are derived from forces\. Operator models are being developed for two main purposes:

![Refer to caption](https://arxiv.org/html/2606.28911v1/Figures/advertisement.png)Figure 1:Applications of electronic structure learning in molecules \(top\) and materials \(bottom\)\. The goal is to learn the mapping between atomic structures \(left\) and their corresponding electronic Hamiltonian matrices \(middle\), and predict them at scales inaccessible by DFT alone\. From these matrices, we can analytically construct electron densities or compute the electronic current flowing through complex materials \(right\)\.1. 1\.Enabling quantum transport through complex materials:The Hamiltonian in matrix form serves as an input to quantum transport \(QT\) simulations that aim to compute the electronic current flowing through atomically\-resolved devices under different voltage configurations\. Advanced QT solvers operate on electronic structure inputs at the level of density functional theory \(DFT\), one of the most powerfulab initiomethods for electronic structure calculations\[[6](https://arxiv.org/html/2606.28911#bib.bib33)\]\. Currently, QT codes are capable of treating systems with 10k\+ atoms, in the presence of complex many\-body effects\[[44](https://arxiv.org/html/2606.28911#bib.bib7)\], matching the physical dimensions of electronic devices fabricated by semiconductor companies\[[1](https://arxiv.org/html/2606.28911#bib.bib152)\]\. However, generating𝐇\\mathbf\{H\}using DFT codes scales with𝒪​\(No​r​b3\)\\mathcal\{O\}\(N\_\{orb\}^\{3\}\),No​r​bN\_\{orb\}being the total number of atomic orbitals in the structure of interest, ultimately limiting the problem sizes \(number of atoms\) that can be handled\. Existing works typically compute𝐇\\mathbf\{H\}for small, periodic systems and tile them to construct device\-scale models\. Meanwhile, these operator models can generate𝐇\\mathbf\{H\}with𝒪​\(No​r​b\)\\mathcal\{O\}\(N\_\{orb\}\)scalability\[[41](https://arxiv.org/html/2606.28911#bib.bib31)\]\. In the same way that MLIPs have enabled new regimes of MD simulations, operator models thus promise to unlock QT modeling of realistically complex atomic geometries\[[47](https://arxiv.org/html/2606.28911#bib.bib38)\]\.
2. 2\.Exploiting scaling laws for downstream properties:ML property models are data\-limited \- their performance improves with more training data\[[46](https://arxiv.org/html/2606.28911#bib.bib10)\]\. However, the forces and energies in existing training datasets have already consumed billions of core hours to be assembled at the DFT level\[[27](https://arxiv.org/html/2606.28911#bib.bib8)\], and generating orders\-of\-magnitude more data is not presently feasible\. Within these datasets, the processing of each molecule or material unit cell ofNNatoms generates𝒪​\(1\)\\mathcal\{O\}\(1\)energy labels,𝒪​\(N\)\\mathcal\{O\}\(N\)force labels, and𝒪​\(N2\)\\mathcal\{O\}\(N^\{2\}\)orbital interaction labels in the electronic structure operators\. Leveraging the latter available but so\-far\-ignored electronic data through pre\-training schemes provides an efficient route to continue exploiting ‘neural scaling laws,’ which define the relationship between training dataset size and model accuracy\. In DFT, the orbital interactions within electronic structure operators parameterize all subsequent atomic\-level molecular and material properties\. Incorporating this information intopropertyprediction architectures improves generalization on atomic\- and molecular\-prediction tasks\[[8](https://arxiv.org/html/2606.28911#bib.bib14),[20](https://arxiv.org/html/2606.28911#bib.bib12),[18](https://arxiv.org/html/2606.28911#bib.bib5),[40](https://arxiv.org/html/2606.28911#bib.bib13),[33](https://arxiv.org/html/2606.28911#bib.bib71)\]\. Small molecule benchmarks have shown that computing forces and energies analytically from learned electronic structure operators leads to accuracy competitive with direct predictions using MLIPs\[[23](https://arxiv.org/html/2606.28911#bib.bib3)\]\.

Sizable datasets of electronic structure matrices have recently become available, with up to 10 PB\[[27](https://arxiv.org/html/2606.28911#bib.bib8)\]of orbital interactions\. Training on this matrix data, however, introduces a different set of computational challenges than previously encountered by ML property models such as MLIPs\. First, the spherical features used within embeddings must match the dimensions of high\-rank orbital interaction data, typically up toff\(and sometimesgg\) orbitals \(requiring angular degreel=6−8l=6\-8\)\. Contrary to MLIPs where the rank of the maximum spherical coefficients is typically a hyperparameter, electronic structure prediction requires high\-rank embeddings simply to produce data of correct dimensions\. Secondly, while MLIPs typically enforce cutoffs of∼\\sim6\-8 Å between interacting atoms, treating orbital interaction data requires handling dedicated edge embeddings that extend to interatomic distances of 12 Å\[[47](https://arxiv.org/html/2606.28911#bib.bib38),[20](https://arxiv.org/html/2606.28911#bib.bib12)\]\. Larger cutoffs create densely connected graph representations with node degrees over 100, significantly increasing the total volume of embeddings to be processed\. Finally, processing orbital interaction data necessitates frequent basis transformations during training and inference to convert between ‘label’ and ‘matrix’ representations of𝐇\\mathbf\{H\}\. Together, these three key differences substantially increase the memory footprint of both training and inference\.

To enable operator prediction ‘at scale’, we therefore introduce theMALOQ\(MassivelyAcceleratedLearning ofOperators forQuantum Transport\) package, an accelerated ML framework to train electronic structure matrices and predict them for arbitrary atomic elements and structure sizes, from large molecules to 10k\+ atom material systems \(Fig\.[1](https://arxiv.org/html/2606.28911#S1.F1)\)\. At the core of MALOQ is a message passing Graph Neural Network \(GNN\) leveraging equivariant spherical channel network \(eSCN\) convolutions\[[37](https://arxiv.org/html/2606.28911#bib.bib62)\]\. Its backbone architecture has achieved state\-of\-the\-art performance for both MLIPs \(eSEN\[[11](https://arxiv.org/html/2606.28911#bib.bib94)\]\) and Hamiltonian prediction tasks \(HELM\[[20](https://arxiv.org/html/2606.28911#bib.bib12)\]\)\. On top of it, we introduce two categories of computational innovations designed to enable massively parallel training and inference of electronic structure matrices:

1. 1\.Operator data processing acceleration:\(1a\)Efficient ‘Matrix↔\\leftrightarrowLabel’ conversions kernels, allowing for Hamiltonian matrices to be pre\-processed into training labels at∼106\\sim 10^\{6\}atoms/s, and rapidly reconstructed from predicted spherical coefficients\.\(1b\)Accelerated computation of the generalized rotation matrices \(Wigner\-D matrices\) required for eSCN convolutions\.
2. 2\.Custom distribution of large molecular and material atomic graphs to accelerate training over molecular datasets, and perform inference for the𝐇\\mathbf\{H\}of large atomic systems\. We combine an efficient point\-to\-point communication scheme with edge\-wise partitioning to minimize communication and reduce load imbalance between partitions, achieving strong scaling up to 192 GH200 GPUs, and\>90%\>90\\%weak scaling efficiency on structure sizes of 12,000 atoms \(∼\\sim10 million orbital interactions\)\.

Together, these contributions enable us to\(1\)improve training speeds over irregular molecular databases by 30\-50%, and\(2\)achieve rapid inference for Hamiltonian matrices on large material graphs containing 10k\+ atoms\. On the one hand, our work allows for the simulation of industry\-relevant atomic systems whose quantum transport properties can now be readily determined at the DFT level with reduced computational complexity\. On the other hand, it enables an efficient treatment of the tremendous volume of molecular and material property data available at electronic resolution, creating an avenue for ML property models to continue benefiting from neural scaling laws without additional dataset generation\.

## IIBackground & Related Work

### II\-AThe electronic structure problem

The training data for MLIPs and other property models is typically generated using DFT\. This first principle method implements the Kohn\-Sham equations, which self\-consistently couple the electron density and electrostatic potential of the atomic system of interest until convergence is reached, afterNi​t​e​rN\_\{iter\}iterations\[[24](https://arxiv.org/html/2606.28911#bib.bib49)\]\. Several commercially or freely available codes implement this self\-consistent and iterative “density↔\\leftrightarrowpotential” scheme on atomic structures\[[26](https://arxiv.org/html/2606.28911#bib.bib149),[32](https://arxiv.org/html/2606.28911#bib.bib130)\]\.

Many of these implementations operate in a basis of localized atomic orbitals\|φi⟩\|\\varphi\_\{i\}\\rangle, often constructed from contracted Gaussian functions\[[26](https://arxiv.org/html/2606.28911#bib.bib149),[32](https://arxiv.org/html/2606.28911#bib.bib130)\]that transform like spherical harmonics \(Yml​\(r^′\)Y^\{l\}\_\{m\}\(\\hat\{r\}^\{\\prime\}\)\) under rotation\. Each spherical harmonic is specified by an angular momentum degreelland orderm∈\{−l,…,l\}m\\in\\\{\-l,\\dots,l\\\}, e\.g\.,s​\(l=0\)s\(l=0\),p​\(l=1\)p\(l=1\),d​\(l=2\)d\(l=2\),⋯\\cdots\. In this basis, the Kohn\-Sham equations take the form of a generalized eigenvalue problem:H​ψ=ε​S​ψ\\textbf\{H\}\\psi=\\varepsilon\\textbf\{S\}\\psi\. The Hamiltonian matrix𝐇\\mathbf\{H\}has entriesHi​j=⟨φi\|H^​\(𝐫\)\|φj⟩H\_\{ij\}=\\langle\\varphi\_\{i\}\|\\hat\{H\}\(\\mathbf\{r\}\)\|\\varphi\_\{j\}\\ranglewhereH^​\(𝐫\)\\hat\{H\}\(\\mathbf\{r\}\)is the Hamiltonian operator\. The overlap matrix𝐒\\mathbf\{S\}is made of termsSi​j=⟨φi\|φj⟩S\_\{ij\}=\\langle\\varphi\_\{i\}\|\\varphi\_\{j\}\\ranglethat describe finite overlaps between the localized orbitals\. They can be pre\-computed from the orbital basis\. Both𝐇\\mathbf\{H\}and𝐒\\mathbf\{S\}are matrices of sizeNo​r​b=∑k=1NANo​r​b,kN\_\{orb\}=\\sum\_\{k=1\}^\{N\_\{A\}\}N\_\{orb,k\}, whereNAN\_\{A\}is the total number of atoms,No​r​b,kN\_\{orb,k\}the number of orbitals \(basis elements\) of atomkk, and the indexkkruns over all atoms accounted for\. The solution of the Kohn\-Sham equations provides the energy eigenvalues \(ε\\varepsilon\) and wavefunctions \(ψ\\psi\) from which the charge densityρ​\(𝐫\)\\rho\(\\mathbf\{r\}\)can be derived\. The core operation within DFT codes then consists of repeatedly solving the aforementioned generalized eigenvalue problem, until self\-consistent convergence is reached betweenρ​\(𝐫\)\\rho\(\\mathbf\{r\}\)and the resulting electrostatic potentialV​\(𝐫\)V\(\\mathbf\{r\}\)\. The computational complexity of the full method scales with𝒪​\(No​r​b3\)\\mathcal\{O\}\(N\_\{orb\}^\{3\}\)per iteration\. At the end of these self\-consistent iterations, atomic\- and molecular\-level properties, such as forces and energies, can be computed from the final, converged𝐇\\mathbf\{H\}\.

### II\-BRelated work

ApplicationOp\.TaskPredictionDistr\.𝐍𝐀\\mathbf\{N\_\{A\}\}\(Lm​a​xL\_\{max\}\)NodeEdgeMLIPs\\multirow@dimawidth 0pt\{\\left\.\\vbox\{\\vrule height=0\.0pt\\multirow@dima width 0pt\}\\textnormal\{\\hbox\{\}\}\\right\\\{DeepMD\[[17](https://arxiv.org/html/2606.28911#bib.bib54)\]LinearF/E✓✗✓\-Allegro\[[25](https://arxiv.org/html/2606.28911#bib.bib58)\]TPF/E✓✗✓\-MACE\[[3](https://arxiv.org/html/2606.28911#bib.bib9)\]TPF/E✓✗✗\-SevenNet\[[36](https://arxiv.org/html/2606.28911#bib.bib36)\]TPF/E✓✗✓\-UMA\[[46](https://arxiv.org/html/2606.28911#bib.bib10)\]eSCNF/E✓✗✓\-Hamiltonians\\multirow@dimawidth 0pt\{\\left\.\\vbox\{\\vrule height=0\.0pt\\multirow@dima width 0pt\}\\textnormal\{\\hbox\{\}\}\\right\\\{QHNet\[[51](https://arxiv.org/html/2606.28911#bib.bib120)\]TPHi​jH\_\{ij\}✓✓✗10 \(4\)WANet\[[30](https://arxiv.org/html/2606.28911#bib.bib11)\]eSCNHi​jH\_\{ij\}✓✓✗100 \(6\)HELM\[[20](https://arxiv.org/html/2606.28911#bib.bib12)\]eSCNHi​jH\_\{ij\}✓✓✗150 \(6\)DeepH\[[29](https://arxiv.org/html/2606.28911#bib.bib37)\]eSCNHi​jH\_\{ij\}✓✓✗216 \(4\)MALOQeSCNHi​jH\_\{ij\}✓✓✓3,000\+ \(4\-8\)TABLE I:Selection of GNN applications for materials modeling across atomic elements \(near\-‘universal’\), including functional similarities and differences in the prediction tasks, operation \(Op\.\) used to update node/edge features, and possibility of distributed compute environment \(Distr\)\. F/E = Forces & Energies,Hi​jH\_\{ij\}= Hamiltonian, TP = SO\(3\) equivariant tensor product, eSCN = SO\(2\) equivariant spherical channel network\.𝐍𝐀\\mathbf\{N\_\{A\}\}= Maximumtrainingstructure size\.GNNs are now mainstream in computational materials science, where they are used to learn structure\-property relationships on atomic graphs\. Training occurs through message passing, where nodes and/or edges of the graph are updated as a learnable function of other nodes/edges \(typically atomic elements/interatomic distance\) from their neighborhoods\. Successive message passing \(MP\) layers gradually transform the initial identity of each node/edge such that they can be mapped to targets representing material properties\.

The success of many GNNs in learning structure\-property relationships hinges on preserving the structural symmetries of the input atomic graphs\. Graph\-based architectures,per se, ensure permutation and translation invariance of atomic coordinates\. Imposing interactions cutoffs allows models to further take advantage of the physical nearsightedness of atomic\-interactions\. Rotational symmetries have been incorporated by explicitly designing the GNNs to commute with rotation operations, typically by using tensor products and gated activations in place of linear layers and nonlinear activation functions\. Compared to otherwise ‘rotationally invariant’ networks, which depend on data\-augmentation to learn these relationships, the ‘rotationally equivariant’ models considered here learn physically\-meaningful mappings with less data\[[4](https://arxiv.org/html/2606.28911#bib.bib41)\]and lower compute budgets\[[7](https://arxiv.org/html/2606.28911#bib.bib42)\]\. While models without encoded rotational symmetries can perform competitively when trained on sufficient property data\[[31](https://arxiv.org/html/2606.28911#bib.bib76)\], for operators such as electronic structure matrices, the data explicitly takes the form of high\-rank embeddings, making incorporation of rotational symmetries unavoidable for sufficient accuracy\[[20](https://arxiv.org/html/2606.28911#bib.bib12),[30](https://arxiv.org/html/2606.28911#bib.bib11)\]\.

We present key features of state\-of\-the\-art GNNs for MLIP and Hamiltonian prediction inTable[I](https://arxiv.org/html/2606.28911#S2.T1), and compare them to MALOQ\. Note that GNNs targeting MLIPs are restricted to ‘node’ prediction tasks, while those dedicated to Hamiltonian matrices must also include ‘edge’ prediction capabilities\. For all models performing electronic structure prediction, we include the maximum system sizes they have so far been reported to achieve\. Two aspects inTable[I](https://arxiv.org/html/2606.28911#S2.T1)are particularly relevant to our work, \(1\) the development of models specific to Hamiltonian prediction and \(2\) domain decomposition/distribution over the underlying atomic graphs\. Below, we discuss their progression and current state of the art\.

DatasetElements𝐍𝐚𝐭𝐨𝐦𝐭𝐨𝐭𝐚𝐥\\mathbf\{N\_\{atom\}^\{total\}\}𝐍𝐚𝐭𝐨𝐦𝐚𝐯𝐠\\mathbf\{N\_\{atom\}^\{avg\}\}𝐝𝐧𝐨𝐝𝐞𝐚𝐯𝐠\\mathbf\{d\_\{node\}^\{avg\}\}Basis\(𝐋max\\mathbf\{L\_\{\\max\}\}\)Func\.rc​u​tr\_\{cut\}\(Å\)SmallmoleculeMD17H, C, N, O\-\-\(FC\)def2\_SVP \(4\)PBE\(FC\)QH9H, C, N, O, F\-\-\(FC\)def2\_SVP \(4\)B3LYP\(FC\)∇2\\nabla^\{2\}DFT \(2k split\)H, C, N, O, F, S, Cl, Br938,92077\(FC\)def2\_SVP \(4\)ω\\omegaB97X\-D\(FC\)LargeMoleculeCSH\_58k×\\times58 elements\-\-\-def2\-TZVPD \(6\)ω\\omegaB97M\-V12\.0Electrolytes×\\times19 elements1,201,3694148def2\-TZVPD \(6\)ω\\omegaB97M\-V12\.0Metal\-Organics×\\times31 elements1,064,6257579def2\-TZVPD \(8\)ω\\omegaB97M\-V12\.0MaterialAmorphous\-HfO2Hf, O9,0003,000598SZV/DZVP \(4\)PBE12\.0Amorphous\-GSTGe, Sb, Te\-\-\-DZVP \(4\)PBE12\.0TABLE II:Summary of existing public Hamiltonian matrix datasets, highlighting the distribution of structure size \(\# atoms,𝐍𝐚𝐭𝐨𝐦𝐭𝐨𝐭𝐚𝐥\\mathbf\{N\_\{atom\}^\{total\}\},𝐍𝐚𝐭𝐨𝐦𝐚𝐯𝐞𝐫𝐚𝐠𝐞\\mathbf\{N\_\{atom\}^\{average\}\}\), node degree \(𝐝𝐧𝐨𝐝𝐞\\mathbf\{d\_\{node\}\}\), basis set and𝐋𝐦𝐚𝐱\\mathbf\{L\_\{max\}\}required to describe it, functional type \(Func\.\), and interaction cutoff radius\. ‘FC’ means ‘fully connected graph’, in case molecules are small enough to avoid applying a cutoff\. Data sources: MD17\[[38](https://arxiv.org/html/2606.28911#bib.bib100)\], QH9\[[50](https://arxiv.org/html/2606.28911#bib.bib28)\],∇2\\nabla^\{2\}DFT\[[22](https://arxiv.org/html/2606.28911#bib.bib4)\], amorphous materials\[[47](https://arxiv.org/html/2606.28911#bib.bib38)\], electronic structure subsets from OMol25\[[27](https://arxiv.org/html/2606.28911#bib.bib8),[20](https://arxiv.org/html/2606.28911#bib.bib12)\]\.#### II\-B1Hamiltonian prediction models

Models which train/predict Hamiltonian matrices were initially constrained to small\-size systems, with most existing implementations treating small\-molecule datasets such as MD17\[[9](https://arxiv.org/html/2606.28911#bib.bib77)\]and QH9\[[50](https://arxiv.org/html/2606.28911#bib.bib28)\], or periodic materials datasets which are described by small, repeated unit cells\[[28](https://arxiv.org/html/2606.28911#bib.bib122)\]\. In these cases, rotational equivariance was satisfied by using tensor products that mix features\[[14](https://arxiv.org/html/2606.28911#bib.bib79),[51](https://arxiv.org/html/2606.28911#bib.bib120)\]\. However, tensor product operations scale poorly withLm​a​xL\_\{max\}, the maximum angular momentum degree of the spherical harmonic basis kept in the model,𝒪​\(Lm​a​x6\)\\mathcal\{O\}\(L\_\{max\}^\{6\}\)when dense, and𝒪​\(Lm​a​x5\)\\mathcal\{O\}\(L\_\{max\}^\{5\}\)when leveraging sparsity in the tensor coefficients\[[48](https://arxiv.org/html/2606.28911#bib.bib75)\]\. Despite significant acceleration with dedicated libraries\[[5](https://arxiv.org/html/2606.28911#bib.bib74)\], the computational complexity of tensor products explode when treating orbital interactions ofLm​a​x≥4L\_\{max\}\\geq 4, as typically encountered in electronic structure matrices\.

Hamiltonian prediction architectures capable of treating systems beyond small molecules have thus taken a different route: They use eSCN convolutions to mix features\. By relaxing strict parity equivariance and implementing local bond rotations to reduce the dimensionality of the spherical interactions from SO\(3\) to SO\(2\), eSCN\-based models enable rotationally equivariant operations built from linear layers, and thus scale with𝒪​\(Lm​a​x3\)\\mathcal\{O\}\(L\_\{max\}^\{3\}\)\[[37](https://arxiv.org/html/2606.28911#bib.bib62)\]\. Currently, all Hamiltonian prediction models capable of treating structures with 100\+ atoms rely on this approach\[[30](https://arxiv.org/html/2606.28911#bib.bib11),[47](https://arxiv.org/html/2606.28911#bib.bib38),[20](https://arxiv.org/html/2606.28911#bib.bib12),[45](https://arxiv.org/html/2606.28911#bib.bib20)\]\. A few of these models have recently been trained on ‘universal’ datasets, making them capable of predicting orbital interactions across a wide range of atomic elements\[[30](https://arxiv.org/html/2606.28911#bib.bib11),[20](https://arxiv.org/html/2606.28911#bib.bib12)\]\. Elsewhere, the scalability of eSCN convolutions has also been leveraged for force/energy prediction, and serves as the backbone operation behind models such as the ‘Universal Model for Atoms’ \(UMA\)\[[46](https://arxiv.org/html/2606.28911#bib.bib10)\]\.

We note that there exists a class of electronic structure prediction models targeting ‘effective’ or ‘empirical tight\-binding’ Hamiltonians rather thanab initioones at the DFT\-level\[[15](https://arxiv.org/html/2606.28911#bib.bib69),[39](https://arxiv.org/html/2606.28911#bib.bib70)\]\. These models typically scale to much larger system sizes, but at the expense of the structural and compositional universality made possible by a DFT\-level basis\.

#### II\-B2Domain decomposition/distribution

The vast majority of distributed GNN applications in materials science are concerned with MLIPs\. Many of them, e\.g\., DeepMD\[[17](https://arxiv.org/html/2606.28911#bib.bib54)\]and Allegro\[[25](https://arxiv.org/html/2606.28911#bib.bib58)\], leverage the domain decomposition functionality of the LAMMPS molecular dynamics code\[[42](https://arxiv.org/html/2606.28911#bib.bib65)\]\. LAMMPS relies on spatial decomposition of the domain into boxes, and has an additional functionality to modify the box\-boundaries and balance the number of atoms within each box\. During message passing iterations, ghost atoms are communicated through halo nodes/edges at the boundaries to aggregate messages onto each node\. The communicated volume is determined by the connectivity of the graph, densely connected graphs incurring high communication overhead as ghost atoms must send messages across multiple layers\.

Directly partitioning the graph instead of the spatial domain appears as a promising approach to improve load balancing and more evenly distribute communication\. This is the approach pursued by SevenNet\[[36](https://arxiv.org/html/2606.28911#bib.bib36),[4](https://arxiv.org/html/2606.28911#bib.bib41)\]and DistMLIP\[[16](https://arxiv.org/html/2606.28911#bib.bib67)\], which serve as wrappers for existing MLIP architectures\. SevenNet is integrated into LAMMPS, and its strong scaling abilities are primarily limited by reduced GPU utilization when distributed over many GPUs\. DistMLIP\[[16](https://arxiv.org/html/2606.28911#bib.bib67)\]supports multi\-GPU \(but single\-node\) inference\. The distribution in these applications generally occurs only over nodes \(atoms\) and not over edges, making them less suitable to Hamiltonian prediction models, where the number of graph edges far exceeds that of nodes\.

### II\-CHamiltonian Matrix Datasets

All DFT\-level datasets available today have necessitated at least one electronic\-structure calculation per atomic system considered, but almost none of them stored the corresponding Hamiltonian matrices\. InTable IIwe summarize the parameters of several datasets that indeed include Hamiltonian matrices\. Note that the nature of the atomic elements as well as the DFT features \(functional, localized basis set\) do not change the computational problem at hand, but heavier elements typically require a larger basis, thus determiningLm​a​xL\_\{max\}\. To benchmark the computational performance of MALOQ when training on large\-scale electronic structure matrix data, we use data from the three representative datasets in bold inTable II: \(1\)∇2\\nabla^\{2\}DFT and its drug\-like molecules with up to 50 atoms each, \(2\) custom ‘unsolvated electrolytes’ and ‘metal organics’ datasets derived from the Open Molecules 2025 4 million data with basis\-dependentLm​a​xL\_\{max\}\[[27](https://arxiv.org/html/2606.28911#bib.bib8)\], and \(3\) the 3,000\-atom/structure amorphous HfO2dataset released by Ref\.\[[47](https://arxiv.org/html/2606.28911#bib.bib38)\]\.

## IIIImplementation and Optimization

MALOQ is an ML model for scalable prediction of electronic\-structure operators, e\.g\., Hamiltonian and density matrices \(rank\-NN\), as well as general operator\-valued quantities in materials modeling, such as Hessians \(rank\-2\)\. We illustrate the stages of MALOQ’s internal workflow during inference inFig\.[2](https://arxiv.org/html/2606.28911#S3.F2)\(a\)\. Overall, it consists of two components: a learnable ML model that learns mappings from atomic structure to spherical tensors, and ‘non\-learnable’ data\-processing pipelines\.

The first ‘non\-learnable’ step, labeled ‘Compute Wigners’, creates generalized rotation \(Wigner\-D\) matrices which rotate all edges of the atomic graph to a common \(\+y\+y\) axis\. These precomputed rotation matrices are then used within the second, learnable component of the model: the eSCN convolutions in each message passing layer\. By rotating the embeddings of each node to a common axis before they interact, eSCN convolutions reduce the dimensionality of the ‘tensor product’ between them from SO\(3\) to SO\(2\)\. The learnable interaction can then be applied through a set of constrained linear layers while preserving rotational equivariance, following a reference implementation first introduced in Ref\.\[[37](https://arxiv.org/html/2606.28911#bib.bib62)\]\. Finally, the directed ‘messages’ created by interacting nodes along each graph edge are either aggregated onto their target nodes \(in the Node blocks\) or used directly to update the embeddings of edges \(in the Edge blocks\)\.

At the end of the message passing operations, each node or edge of the atomic graph is described by an embedding which has been processed through three layers of learnable eSCN convolutions\. At this point, the final ‘non\-learnable’ component of MALOQ \(’Label→\\rightarrowMatrix inFig\.[2](https://arxiv.org/html/2606.28911#S3.F2)\(a\)\) uses these predicted ‘Labels’ to reconstruct the Hamiltonian in matrix representation\. Note that during training, the opposite transformation is applied to the Hamiltonian matrices within the training dataset: ’Matrix→\\rightarrowLabel performs basis transformations between the original, matrix representation of𝐇\\mathbf\{H\}, and a set of labels which can be mapped to/learned by the individual nodes and edges of a GNN corresponding to the atomic structure of interest\.

The non\-learnable ‘data pipelines’ within MALOQ have been accelerated through custom kernels that enable processing \(training\) over large dataset batches and inference for large system sizes\. These large data volumes naturally exceed GPU memory constraints, necessitating a graph\-level parallelization to process the corresponding embeddings through layers of eSCN convolutions\. We thus additionally introduce a distributed\-graph implementation which efficiently communicates embeddings across multiple processes, and illustrate partitioning strategies that reduce load imbalance across batches of molecules with diverse size, and large, single\-materials graphs\. Together, these computational contributions enable both rapid, scalable operator prediction, bringing the accuracy of MALOQ’s backbone architecture\[[20](https://arxiv.org/html/2606.28911#bib.bib12)\]to 10k\+ atom systems\. In this section, we detail each of these components, starting with the methods behind them, associated computational challenges, and how they are optimized within the MALOQ code\. We close each subsection with a brief evaluation of the computational improvements achieved\.

![Refer to caption](https://arxiv.org/html/2606.28911v1/Figures/matrix_label.png)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→\\rightarrowLabel’ conversion occurs\. \(b\) Illustration showing the mapping between the matrix and label representations of data for a toy H2O molecule with a minimal basis \(Oxygen:ss,pp, Hydrogen:ss\), so thatLm​a​xL\_\{max\}=2\. The labels for each node/edge are populated with diagonal/off\-diagonal submatrices\. Measured processing throughput/runtime to either \(c\) convert multiple matrices to labels, or \(d\) reconstruct a single matrix from a given set of predicted data\. All measurements were taken over a subset of molecules present in the∇2\\nabla^\{2\}DFT \(Lm​a​xL\_\{max\}=4\) and OMol Electrolytes/Metal\-Organics \(Lm​a​xL\_\{max\}=6/8\) datasets, with data processed in fp32\.### III\-ATraining Environment

We first introduce our experimental setup\. Our models are trained on the Alps supercomputer at CSCS, where each node comprises 4 NVIDIA GH200 Grace Hopper superchips\[[12](https://arxiv.org/html/2606.28911#bib.bib137)\]\. Every superchip combines a Hopper GPU with 132 Streaming Multiprocessors and 96 GB HBM3 memory with a Grace CPU with 72 cores and 128 GB LPDDR5\. Each Hopper GPU is connected to every other GPU within the same node via NVLink and 150 GB/s bidirectional point\-to\-point bandwidth, resulting in a total of 900 GB/s\. The Alps nodes are connected via an HPE Slingshot 11 network with a Dragonfly topology\. MALOQ is written in PyTorch\[[2](https://arxiv.org/html/2606.28911#bib.bib140)\], using torch tensors to represent the embeddings of nodes and edges\. To communicate embeddings between GPUs, we use PyTorch’s distributed functionality and its NVIDIA Collective Communications Library \(NCCL\) backend\.

### III\-BData pipelines: Matrix↔\\leftrightarrowLabel conversions

#### III\-B1Methods

The entries of𝐇\\mathbf\{H\}correspond to interactions between electronic orbitals located on the same or on different atoms\. The full matrix can be decomposed into sub\-matrices𝐇i​jα​β=⟨lα,mα\|​𝐇​\|lβ,mβ⟩\{\\mathbf\{H\}^\{\\alpha\\beta\}\_\{ij\}\}=\\bra\{l\_\{\\alpha\},m\_\{\\alpha\}\}\\mathbf\{H\}\\ket\{l\_\{\\beta\},m\_\{\\beta\}\}\. They contain all interactions𝐇i​jα​β​\(mα,mβ\)\{\\mathbf\{H\}^\{\\alpha\\beta\}\_\{ij\}\}\(m\_\{\\alpha\},m\_\{\\beta\}\)between spherical harmonic basis functionslαl\_\{\\alpha\}\(of degreeα\\alpha\) on atomii, andlβl\_\{\\beta\}\(of degreeβ\\beta\) onjj, as illustrated in the left part ofFig\.[2](https://arxiv.org/html/2606.28911#S3.F2)\(b\)\. Mathematically, each𝐇i​jα​β\{\\mathbf\{H\}^\{\\alpha\\beta\}\_\{ij\}\}corresponds to the tensor productlα⊗lβl\_\{\\alpha\}\\otimes l\_\{\\beta\}, which has length\(2​lα\+1\)×\(2​lβ\+1\)\(2l\_\{\\alpha\}\+1\)\\times\(2l\_\{\\beta\}\+1\)\. Through a transformationTT, this tensor product can be written as the direct sum \(⊕\\oplus\) of angular momentum eigenstates\|L,M⟩\\ket\{L,M\}, whereLLranges from degree\|α−β\|\|\\alpha\-\\beta\|to\|α\+β\|\|\\alpha\+\\beta\|, i\.e\.,T​\(lα⊗lβ\)=⨁i=\|α−β\|\(α\+β\)Li=L\|α−β\|⊕…⊕L\(α\+β\)T\(l\_\{\\alpha\}\\otimes l\_\{\\beta\}\)=\\bigoplus\_\{i=\|\\alpha\-\\beta\|\}^\{\(\\alpha\+\\beta\)\}L\_\{i\}=L\_\{\|\\alpha\-\\beta\|\}\\oplus\.\.\.\\oplus L\_\{\(\\alpha\+\\beta\)\}\. The transformationTTinvolves a matrix of Clebsch\-Gordon \(CG\) coefficients\. The entries for a specific\|L,M⟩\\ket\{L,M\}component are given by:

\|L,M⟩=∑mα=−lα\+lα∑mβ=−lβ\+lβC\(lα,mα\)​\(lβ,mβ\)\(L,M\)​\|lα,mα⟩​\|lβ,mβ⟩\.\\ket\{L,M\}=\\sum\_\{m\_\{\\alpha\}=\-l\_\{\\alpha\}\}^\{\+l\_\{\\alpha\}\}\\sum\_\{m\_\{\\beta\}=\-l\_\{\\beta\}\}^\{\+l\_\{\\beta\}\}C\_\{\(l\_\{\\alpha\},m\_\{\\alpha\}\)\(l\_\{\\beta\},m\_\{\\beta\}\)\}^\{\(L,M\)\}\\ket\{l\_\{\\alpha\},m\_\{\\alpha\}\}\\ket\{l\_\{\\beta\},m\_\{\\beta\}\}\.\(1\)
Here,C\(lα,mα\)​\(lβ,mβ\)\(L,M\)C\_\{\(l\_\{\\alpha\},m\_\{\\alpha\}\)\(l\_\{\\beta\},m\_\{\\beta\}\)\}^\{\(L,M\)\}is the CG coefficient describing the contribution of\|lα,mα⟩​\|lβ,mβ⟩\\ket\{l\_\{\\alpha\},m\_\{\\alpha\}\}\\ket\{l\_\{\\beta\},m\_\{\\beta\}\}to the state\|L,M⟩\\ket\{L,M\}, and\|lα,mα⟩​\|lβ,mβ⟩\\ket\{l\_\{\\alpha\},m\_\{\\alpha\}\}\\ket\{l\_\{\\beta\},m\_\{\\beta\}\}corresponds to𝐇i​jα​β​\(mα,mβ\)\{\\mathbf\{H\}^\{\\alpha\\beta\}\_\{ij\}\}\(m\_\{\\alpha\},m\_\{\\beta\}\)\. The transformationTTthus decomposes the Hamiltonian sub\-matrix𝐇i​jα​β\\smash\{\\mathbf\{H\}^\{\\alpha\\beta\}\_\{ij\}\}into a ‘vector’ form, representing a direct sum of tensors with different angular degreell\. All\|L,M⟩\\ket\{L,M\}corresponding to a single𝐇i​j\{\\mathbf\{H\}\_\{ij\}\}block are then concatenated into a vector𝐡i​j\\mathbf\{h\}\_\{ij\}\. The same operation is repeated for all blocks of𝐇\\mathbf\{H\}\. Since not all atomic elements require the same number of orbitals \(basis set\) to be accurately described, the resulting𝐡i​j\\mathbf\{h\}\_\{ij\}vectors have different lengths\. They are all padded to a common dimension, that of the largest element in the dataset\. The collection of all padded𝐡i​j\\mathbf\{h\}\_\{ij\}forms the ‘labels’ used to train the node and edge embeddings of the GNN \(right part ofFig\.[2](https://arxiv.org/html/2606.28911#S3.F2)\(b\)\), and the loss is computed element\-wise over the model’s prediction of them\. The maximum degree of the𝐡i​j\\mathbf\{h\}\_\{ij\}becomes theLm​a​xL\_\{max\}required for embeddings\.

#### III\-B2Optimization

To process a dataset of Hamiltonian matrices, we need to \(1\) extract the individual orbital interaction blocks into equally\-shaped buffers and \(2\) perform a basis transformation to convert them into labels usingEq\. \([1](https://arxiv.org/html/2606.28911#S3.E1)\)\. While the matrix operations within the basis transformation step can be done efficiently using PyTorch tensors, the coefficient extraction inFig\.[2](https://arxiv.org/html/2606.28911#S3.F2)\(b\)rapidly becomes a bottleneck for larger training datasets or inference system sizes\.

We introduce a dedicated CUDA kernel to handle the transformations between batches of \(or single\) matrices and labels\. Within this kernel, each thread processes a single matrix element, i\.e\., orbital\-orbital interaction block, and each thread block processes one node \(𝐇i​iα​β\{\\mathbf\{H\}^\{\\alpha\\beta\}\_\{ii\}\}\) or edge \(𝐇i​jα​β\{\\mathbf\{H\}^\{\\alpha\\beta\}\_\{ij\}\}\)\. The kernel uses an ‘orbital template’𝒯​\(Zi,Zj\)\\mathcal\{T\}\(Z\_\{i\},Z\_\{j\}\)containing the row and column slices of𝐇i​j\\mathbf\{H\}\_\{ij\}corresponding to the set of orbitals interaction between atomiiof atomic elementZiZ\_\{i\}and atomjjof atomic elementZjZ\_\{j\}\. This information can be precomputed for a given basis\.

As the size of the labels differs for each dataset, we benchmark our kernel on a selection of three molecular datasets inTable[II](https://arxiv.org/html/2606.28911#S2.T2), each with differentLm​a​xL\_\{max\}and orbital basis\. To evaluate the performance under realistic conditions, we consider two different scenarios: \(i\) data processing \(forward transformation, pre\-training\) and \(ii\) matrix reconstruction \(backward transformation, post\-inference\)\. First,Fig\.[2](https://arxiv.org/html/2606.28911#S3.F2)\(c\)reports the ‘data processing’ case \(‘Matrices→\\rightarrowLabels’\), where multiple Hamiltonian matrices𝐇\\mathbf\{H\}are converted into orbital interaction labels𝐡\\mathbf\{h\}for supervised learning\. We evaluate processing throughput by considering increasing fractions of each of the three datasets, resulting in increased total \# atoms\. Processing throughput between 7\.3×\\times105and 1\.2×\\times106atoms/s are measured, allowing us to process the largest molecular datasets of 110M structures\[[27](https://arxiv.org/html/2606.28911#bib.bib8)\]\(∼\\sim50 atoms/structure\) for training in under an hour, on a single process\. Note that forLm​a​x=8L\_\{max\}=8, the data exceeds the memory of a single GH200 GPU before peak processing throughput can be reached\. We therefore report the maximum value attained\. Secondly, inFig\.[2](https://arxiv.org/html/2606.28911#S3.F2)\(d\), the ‘matrix reconstruction‘ case is presented \(‘Labels→\\rightarrowMatrix’,𝐡→𝐇\\mathbf\{h\}\\rightarrow\\mathbf\{H\}\)\. At the end of the inference step, the model generates orbital interaction labels𝐡\\mathbf\{h\}for a given atomic structure, which are then re\-assembled into the𝐇\\mathbf\{H\}for downstream applications\. Here, the difference in runtime stems primarily from the number of elements per label \(Lm​a​xL\_\{max\}\), but remains under 10 ms across a range of system sizes\. Note that the variability in measured time/throughput at the same \# atoms can be attributed to different molecular connectivity \(\# edges per atomic graph\)\.

![Refer to caption](https://arxiv.org/html/2606.28911v1/Figures/wigner_d_1line.png)Figure 3:Fused Triton kernel for Wigner\-DDmatrix construction\.\(a\) 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 matrices rotate the node/edge embeddingni/ei​jn\_\{i\}/e\_\{ij\}of every edger^i​j\\hat\{r\}\_\{ij\}from the atomic graph to align it with the\+y\+y\-axis\. \(c\) Doing this requires first computing the Euler angles \(α,β,γ\\alpha,\\beta,\\gamma\) of the rotation from edge vectorsr^i​j\\hat\{r\}\_\{ij\}, and then constructingD\(ℓ\)D^\{\(\\ell\)\}via four tl\.dot operations, which we encapsulate into a fused Triton kernel\. \(d\) Block\-packing strategy for matrix product \(Triton tl\.dot\):ℓ\\ell\-blocks are packed into16×1616\{\\times\}16tiles \(or32×3232\{\\times\}32forℓ=8\\ell\{=\}8\) to maximize tensor core utilization\. \(e\) Latency comparison between our Triton kernel and the Torch implementation at selected number of input edges\.

### III\-CData pipelines: Accelerated Wigner\-DDmatrix generation

#### III\-C1Methods

We now turn to the Wigner\-D matrix block inFig\.[3](https://arxiv.org/html/2606.28911#S3.F3)\(a\)\. For a given input atomic graph, MALOQ first pre\-computes the set of generalized rotation matricesRRthat rotate the graph edges \(ei​je\_\{ij\}\) to a common \(\+y\+y\) axis \(Fig\.[3](https://arxiv.org/html/2606.28911#S3.F3)\(b\), top\)\. As in many equivariant GNNs for materials science, the embeddings of every node \(nin\_\{i\}\) and edge \(ei​je\_\{ij\}\) often take the structure of spherical harmonic coefficients of rankL=0​…​Lm​a​xL=0\.\.\.L\_\{max\}\. EachLL\-coefficient is expanded intoEEchannels \- the dimensions of every node \(nin\_\{i\}\) and edge \(ei​je\_\{ij\}\) embedding processed is thus\(Lm​a​x\+1\)2×E\(L\_\{max\}\+1\)^\{2\}\\times E\. Performing a rotationRRof the original atomic systemr^\\hat\{r\}intor^′=R⋅r^\\hat\{r\}^\{\\prime\}=R\\cdot\\hat\{r\}\(wherer^\\hat\{r\}/r^′\\hat\{r\}^\{\\prime\}are normalized direction vectors\) requires the embeddingsnin\_\{i\}/ei​je\_\{ij\}to be multiplied by a direct sum of ‘Wigner\-D’ matrixDL​\(R\)D^\{L\}\(R\)of degreeLL\., i\.e\.,ni′n\_\{i\}^\{\\prime\}/ei​j′=⨁l=0Lm​a​xDl​\(R\)×nie\_\{ij\}^\{\\prime\}=\\bigoplus\_\{l=0\}^\{L\_\{max\}\}D^\{l\}\(R\)\\times n\_\{i\}/ei​jle\_\{ij\}^\{l\}\. The lower part ofFig\.[3](https://arxiv.org/html/2606.28911#S3.F3)\(b\)illustrates the block\-diagonal structure of these Wigner\-D matrices which rotate the corresponding embeddingsnin\_\{i\}/ei​je\_\{ij\}\. Practically, creating the Wigner\-D matrices involves \(1\) computing a set of Euler anglesα,β,γ\\alpha,\\beta,\\gammacorresponding to each edge in a given atomic graph, and then \(2\) using them to determine the \(2LL\+1\)×\(\)\\times\(2LL\+1\) entries ofDL​\(R\)D^\{L\}\(R\)\.

#### III\-C2Optimization

A reference PyTorch implementation is provided by the e3nn library\[[13](https://arxiv.org/html/2606.28911#bib.bib30)\], which computes eachD​\(R\)LD\(R\)^\{L\}submatrix separately, but for all graph edges in a batch\. Each sub\-computation involves a chain of four small matrix multiplications, executed via separate kernel launches, materializing the intermediate results in GPU global memory\. Thus, a significant speedup can be gained by developing a fused kernel to perform the fullr^i​j\\hat\{r\}\_\{ij\}→\\rightarrowD​\(R\)LD\(R\)^\{L\}operation chain for every graph edger^i​j\\hat\{r\}\_\{ij\}and orbital degreell\(Fig\.[3](https://arxiv.org/html/2606.28911#S3.F3)\(c\)\), vastly increasing arithmetic intensity\. This kernel, which we implement in Triton, \(1\) fuses the calculation of Euler angles and of all matrix products that are required to obtain the Wigner\-D sub\-matrices for every degreeL​…​Lm​a​xL\.\.\.L\_\{max\}, and \(2\) implements a block\-packing strategy to group differentLLs \(Fig\.[3](https://arxiv.org/html/2606.28911#S3.F3)\(d\)\)\. For the latter, we take advantage of the regular angular dimensions of eachLL, which are fixed by the underlying physics: The size of each sub\-matrixDlD^\{l\}is \(2ll\+1\)×\(\\times\(2ll\+1\), to account for the coefficientsm=−l​…\+lm=\-l\.\.\.\+lfor a spherical harmonic of degreell\. As the matrix product in Triton \(tl\.dot\) uses fixed shapes \(e\.g\., 16×\\times16, 32×\\times32…\), packing the sub\-matrices for multipleLL\-components together minimizes padding overhead and fully utilizes the fixed\-size tiles\.

We benchmark the reference PyTorch and our Triton implementations with three differentLm​a​xL\_\{max\}values, corresponding to selected datasets fromTable[II](https://arxiv.org/html/2606.28911#S2.T2)\. The results are presented inFig\.[3](https://arxiv.org/html/2606.28911#S3.F3)\(e\)\. At smaller data sizes \(number of edges, andLm​a​xL\_\{max\}\), the execution time in the PyTorch reference implementation is dominated by kernel launch overhead from multiple matrix products\. In this range, our Triton code shows speedups of∼\\sim500×\\times\(100\-1K edges\) to∼\\sim50×\\times\(10K edges\), as the launching overhead was removed through kernel fusion\. At higher data volumes \(100K\+ edges\), the reference implementation becomes limited by global memory traffic\. Still, our Triton implementation achieves a∼\\sim5\-6×\\timesspeedup from the combination of kernel fusion andLL\-wise block packing\.

![Refer to caption](https://arxiv.org/html/2606.28911v1/Figures/dist.png)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 messagesmi→jm\_\{i\\rightarrow j\}within a toy H2O molecule\. \(c\) Illustration of the send/recv operation implementation to communicate node embeddings from Rank 2 to Ranks 0 and 1 for the example partitioned graph pictured in the inset\. The arrows indicate the edges starting from nodes belonging to Rank 2’s partition and ending in one of the other partitions: 4 \(5\) Rank 2’s nodes are connected to Rank 0 \(1\)’s nodes through inter\-partition edges\. The corresponding data is first copied to a buffer before being sent to the targeted Rank\. There, it is unpacked into a local message buffer\.

### III\-DGraph level distribution

At the start of each MP layer \(indicated with black frames inFig\.[4](https://arxiv.org/html/2606.28911#S3.F4)\(a\)\), a set of messagesmi→jm\_\{i\\rightarrow j\}, consisting of the concatenated node embeddingsni⊕njn\_\{i\}\\oplus n\_\{j\}, are assembled\.Figure[4](https://arxiv.org/html/2606.28911#S3.F4)\(b\)illustrates this process for a fully\-connected toy molecule of 3 atoms, picturing the construction of each of the 6 messages required\. This set of messages undergoes eSCN convolutions, which mix the features ofnin\_\{i\}andnjn\_\{j\}, and are then either aggregated onto nodejj\(node update\) or used to update the edge embeddingei​je\_\{ij\}\(edge update\)\. In a distributed\-memory setting, constructing these messages requires communication of node embeddings across graph partitions\. Our implementation uses PyTorch’s distributed NCCL backend and aggregates the communication required between any two processes as follows: Each process \(1\) packs its ownnin\_\{i\}into a separate send buffer for every one of its neighbors \(jj\), \(2\) sends/receives those buffers using point\-to\-point communication, and \(3\) unpacks the receivednin\_\{i\}and inserts them into its message tensors\. This communication scheme is visualized inFig\.[4](https://arxiv.org/html/2606.28911#S3.F4)\(c\), where a process \(Rank 2\) packs and sends its localnin\_\{i\}required by its neighbors \(Ranks 0 and 1\), which receive and insert them into theirmi→jm\_\{i\\rightarrow j\}\. During the backward pass, the reverse communication pattern is executed, implemented as a custom function in PyTorch\.

In addition, we make several design choices to further refine the communication step\. First, we adopt a target\-nodes\-own\-edges distribution policy: Each rank owns the edgesei​je\_\{ij\}where the connected nodejjis local, eliminating additional inter\-process communication during the edge update phase\. In addition, we partially hide communication latency by overlapping the indexing of local nodes into message tensors with the asynchronous transfer of remote node embeddings\. To that end, we partition the list of local node embeddings such that those requiring transmission to remote ranks are contiguous in memory, as shown for Rank 2 inFig\.[4](https://arxiv.org/html/2606.28911#S3.F4)\(c\)\.

The remaining objective is to partition the graph in order to evenly distribute the workload and reduce communication time\. In the following, we discuss and compare the different graph partitioning approaches implemented in MALOQ for applications across molecules and materials\.

![Refer to caption](https://arxiv.org/html/2606.28911v1/Figures/omol_dist_32rank.png)Figure 5:Training runtime comparison under different distribution schemes\.\(a\) Dataset used for training\. It consists of 128 randomly\-selected molecules from the OMol electrolytes dataset and their corresponding𝐇\\mathbf\{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 molecules are divided across 32 ranks \(molecule\-wise\)\. \(c\) Forward and \(d\) backward pass time distribution across different ranks\. Each data is the median of 200 training epochs, of which the first 20 are discarded\. The vertical dashed lines indicate the maximum time for each distribution scheme\.#### III\-D1Partitioning molecular datasets

Training on molecular datasets in a distributed\-memory environment typically involves creating a molecule\-wise partition, where each rank processes a subset of molecules\. Load imbalance might arise if these molecules are made of a wide range of atoms, which may contain an even wider distribution of connectivity \(\# edges per structure\)\. This load imbalance can be reduced by ‘binning’ molecules into partitions to balance the total number of atoms/edges across the structures handled by each process\. However, a finer load balance can be achieved by partitioning the molecules at the atom/edge level\. In practice, this can significantly reduce the time to process each training batch\.

![Refer to caption](https://arxiv.org/html/2606.28911v1/Figures/edge_throughput.png)Figure 6:Embedding communication and processing throughput\. \(a\) Measured intra\-/inter\-node communication throughput of node embeddingsnin\_\{i\}with embedding sizeEE=128\. \(b\) Processing throughput for eSCN convolutions as a function of the number of messagesmi→jm\_\{i\\rightarrow j\}for differentLm​a​xL\_\{max\}\. Each data point corresponds to the median of 20 measurements\.![Refer to caption](https://arxiv.org/html/2606.28911v1/Figures/molecule_strongscaling.png)Figure 7:Strong scaling of MALOQ over three irregular molecular datasets from Table[II](https://arxiv.org/html/2606.28911#S2.T2)with differentLm​a​xL\_\{max\}\.\(a\) 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 runtime across ranks\. Each data point is annotated by the average number of edges owned by each process under that distribution, to compare with the saturation of embedding processing throughput measured inFig\.[6](https://arxiv.org/html/2606.28911#S3.F6)\. Note that we start from the minimum number of GPUs allowing for the partitioned batch to fit in memory\.As an example, we consider inFig\.[5](https://arxiv.org/html/2606.28911#S3.F5)a batch of 128 molecules from the OMol\-electrolytes dataset \(Fig\.[5](https://arxiv.org/html/2606.28911#S3.F5)\(a\)\) and compare the workload distribution \(Fig\.[5](https://arxiv.org/html/2606.28911#S3.F5)\(b\)\), as well as the runtime per forward \(Fig\.[5](https://arxiv.org/html/2606.28911#S3.F5)\(c\)\) and backward pass \(Fig\.[5](https://arxiv.org/html/2606.28911#S3.F5)\(d\)\) under four different partitioning schemes\. The so\-called ‘Metis’ approach uses the METIS package\[[21](https://arxiv.org/html/2606.28911#bib.bib96)\]to search for a mincut solution to the partitioning problem\. Since cuts to the graph are penalized, Metis finds an ‘optimal binning’ when there are more molecules than ranks, thus balancing the partitions without introducing inter\-process communication\. This is the case inFig\.[5](https://arxiv.org/html/2606.28911#S3.F5)\(b\)\. Meanwhile, in the ‘atom\-wise’/‘edge\-wise’ partitions, nodes/edges are evenly distributed among ranks, starting from a list of atoms across the data batch where those within each molecule are contiguous\. In particular, an edge\-wise distribution of the graph leads to a highly balanced embedding workload\. Compared to the molecule\-wise distribution, splitting the batch edge\-wise reduces the time per forward pass by 30% \(from 0\.23 to 0\.15s\), and the time per backwards pass by over 50% \(0\.85 to 0\.4s\), despite the introduction of inter\-process communication\.

#### III\-D2Partitioning large materials graphs

In case of large atomic structures, the corresponding materials graph must be distributed over multiple processes\. The resulting communication involves only a few embeddings exchanged between any two ranks, and is eventually limited entirely by overhead\. To highlight this effect, we show inFig\.[6](https://arxiv.org/html/2606.28911#S3.F6)\(a\)the time to send/receive different counts ofnin\_\{i\}between two processes, each running on a separate GH200 superchip on Alps\. Up to about 100nin\_\{i\}\(consideringE=128E=128channels andLm​a​x=4L\_\{max\}=4\), communication is bound by latency\. At the same time, the achievable runtime improvements due to graph distribution are limited by processing throughput, which decreases once each partition owns too few local edges\. The number of messages required to saturate processing throughput is shown inFig\.[6](https://arxiv.org/html/2606.28911#S3.F6)\(b\)\. It is proportional to the number of local edges\.

To minimize the number of partition neighbors and data exchanges in large materials graphs \(and thus the latency\), we introduce an additional partitioning method based on recursive bisection\. Our ‘modified recursive bisection’ scheme recursively distributes a graph among 2nranks while maintaining edge\-wise balanced partitions, accounting for 3\-D periodicity used in DFT calculations of materials, and minimizing the number of new neighbors introduced by each local cut\. Starting from standard recursive bisection, we make the following changes:\(1\)The dimension along which to cut is determined based on ‘which dimension would create the fewest additional neighboring partitions when cut’, considering both \(a\) the interaction cutoff radiusrc​u​tr\_\{cut\}and \(b\) periodicity of the 3\-D domain\.\(2\)Each sub\-domain, at any stage of the algorithm, is partitioned so that the total node degree \(ND\), which we use as a proxy for the number of edges, remains balanced between the two newly formed partitions\.

## IVApplications

MALOQ is designed to\(1\)improve parallel efficiency \(time per epoch\) when training on massive \(terabyte\-scale\) Hamiltonian matrix datasets and\(2\)rapidly infer electronic structure matrices for materials at scale\. In this section, we present performance benchmarks for real training scenarios under both operating conditions, using the datasets fromTable II\. These benchmarks are specific to the hardware discussed inSection III\-A\- other architectures with different memory constraints are likely to show different scaling performance\.

### IV\-AScalable training over large molecular datasets

‘Universal’ molecular datasets with diverse atomic elements typically necessitate large batch sizes during training, to ensure sufficient sampling over all of the represented elements within each batch\. We thus standardize all our benchmarks using a single batch size of 128 randomly selected molecules, which is distributed across multiple processes\. In all cases, we use an edge\-wise partitioning, as previously explored inFig\.[5](https://arxiv.org/html/2606.28911#S3.F5)\. Note that tackling the largest molecules \(in terms of \# atoms,Lm​a​xL\_\{max\}\) across these datasets already requires partitioning individual molecular graphs to fit in GPU memory\.

The strong scaling results for the training runtime of three molecular datasets are plotted inFig\.[7](https://arxiv.org/html/2606.28911#S3.F7):∇2\\nabla^\{2\}DFT, electrolytes, and metal\-organics \(Table[II](https://arxiv.org/html/2606.28911#S2.T2)\)\. Each dataset necessitates a differentLm​a​xL\_\{max\}, up to 8 for metal\-organics which contain lanthanide elements withgg\-orbitals\. We train these datasets on up to 32 Alps nodes \(128 GH200 superchips\), where the minimum number of nodes depends on the dataset:∇2\\nabla^\{2\}DFT’s batch fits on a single node, electrolytes require at least 4 nodes, and metal\-organics cannot run on less than 12 nodes\. Particularly in the forward pass, the number of edges per GPU at which the scaling efficiency starts dropping roughly corresponds to the point where processing throughput is lost inFig\.[6](https://arxiv.org/html/2606.28911#S3.F6)\(b\):∼\\sim10k for∇2\\nabla^\{2\}DFT withLm​a​x=4L\_\{max\}=4, and∼\\sim5k for the electrolytes withLm​a​x=6L\_\{max\}=6\. The times for the metal\-organics molecules scale well up to 32 nodes, since the average number of edges per partition \(6\.1k\) remains well above the low saturation point forLm​a​x=8L\_\{max\}=8\(about 2k\)\.

### IV\-BTraining and inference over large materials

![Refer to caption](https://arxiv.org/html/2606.28911v1/Figures/hfo2_strong_weak_scaling.png)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 HfO2structure \(shown in inset\)\. \(b\) Weak scaling over inference for the same structure as in \(a\)\. The problem size is increased according to two tiling scheme: 1\-D \(tiling alongxx\) and 2\-D \(tiling alongyyandzz\)\. We measure 92\.4% scaling efficiency between 64 and 256 GPUs in the former case, 96\.4% in the latter\. For \(a\-b\), the reported measurements represent the per\-rank median over 180 epochs, after 20 warm\-up epochs\. Each data point also contains a range \(fill\-between\) indicating the variability in runtime across ranks\.Describing material phenomena like phase boundaries\[[53](https://arxiv.org/html/2606.28911#bib.bib103)\], interfaces between different materials\[[10](https://arxiv.org/html/2606.28911#bib.bib101)\], or amorphous structures\[[47](https://arxiv.org/html/2606.28911#bib.bib38)\], translates to the construction of large material graphs containing 10k\+ atoms\. When processing node and edge embeddings for structures at these scales, taking advantage of the memory of multiple GPUs through graph\-level distribution becomes necessary\.

First, we test the effect of different partitioning schemes when training on large material graphs\. We report inFig\.[8](https://arxiv.org/html/2606.28911#S4.F8)\(a\)the strong scaling performance of MALOQ for the HfO2material dataset fromTable II\. Here, a single structure, corresponding to one batch, contains 3,000 atoms, and roughly 1\.8 million edges under a cutoffrc​u​t=6\.0r\_\{cut\}=6\.0Å\. We measure strong scaling efficiency up to 512 GPUs \(128 nodes\) on the Alps supercomputer \- at this distribution, each rank has, on average∼3×103\\sim 3\\times 10^\{3\}edges\. This is the maximum ‘reasonable’ distribution for this problem size, as further distribution would reduce the number of edges per rank, with loss of processing throughput as a consequence \(seeFig\.[6](https://arxiv.org/html/2606.28911#S3.F6)\(b\)\)\. We compare three partitioning strategies:\(1\)the same edge\-wise distribution as for molecules, now performed on the material graph sorted byxx\-coordinates,\(2\)the Metis algorithm, and\(3\)our modified recursive bisection\. Partitioning with Metis significantly improves strong scaling over a straightforward edge\-wise distribution, as it preserves clusters of locality in such 3\-D material graphs\. The modified recursive bisection approach slightly outperforms Metis on 512 GPUs, where significant communication times amplify the impact of the partitioning method\. In general, Metis and our recursive bisection approach both enable scaling up to 512 GPUs for the backward pass and up to 192 GPUs for the forward pass\.

Figure[8](https://arxiv.org/html/2606.28911#S4.F8)\(b\)presents weak scaling efficiency for inference using the same amorphous\-HfO2system\. We start on 16 Alps nodes \(64 GH200 superchips\) with the baseline structure made of 3,000 atoms \(∼\\sim1\.8 million edges\), and scale up to 64 nodes and 12,000 atoms \(∼\\sim10 million edges\)\. To reliably increase the problem size, we tile the initial 3,000\-atom HfO2cell according to two strategies:\(1\)1\-D tiling, where the cell is repeated only along thexx\-direction, and\(2\)2\-D tiling, where the cell is repeated along theyy\- andzz\-dimensions\. In both cases, we use our modified recursive bisection algorithm to partition the tiled graphs\. MALOQ achieves nearly ideal weak\-scaling efficiency for inference on these structures, 94% with 1\-D scaling and 98% with 2\-D scaling\.

## VConclusion and Outlook

We introduced MALOQ, a scalable, distributed ML operator prediction code capable of treating higher\-rank tensor properties, such as Hessians \(rank\-2\)\[[8](https://arxiv.org/html/2606.28911#bib.bib14)\], and electronic structure \(Hamiltonian/Density\) matrices where the rank \(Lm​a​xL\_\{max\}\) is determined by the orbital basis of the underlying data\[[20](https://arxiv.org/html/2606.28911#bib.bib12),[30](https://arxiv.org/html/2606.28911#bib.bib11),[34](https://arxiv.org/html/2606.28911#bib.bib110),[14](https://arxiv.org/html/2606.28911#bib.bib79)\]\. It shares its backbone architecture \(distributed equivariant GNN with eSCN convolutions\) with common energy/force property prediction models, thus allowing for distributed pre\-training and finetuning across different quantities\. MALOQ can be combined with advanced quantum transport solvers to reveal key features of electronic devices requiring large atomic structures, such as phase\-change memory\[[53](https://arxiv.org/html/2606.28911#bib.bib103)\], silicon nano\-ribbon field\-effect transistors\[[49](https://arxiv.org/html/2606.28911#bib.bib34)\], or nano\-ionic resistive memories\[[35](https://arxiv.org/html/2606.28911#bib.bib102),[19](https://arxiv.org/html/2606.28911#bib.bib88)\]\.

## Acknowledgment

This work was supported by the Swiss National Science Foundation \(SNSF\) under grantn∘\\mathrm\{n^\{\\circ\}\}209358 \(QuaTrEx\), and by the Platform for Advanced Scientific Computing in Switzerland \(BoostQT\)\. We acknowledge support from CSCS \(projects c33, lp16, lp82\)\. The authors would like to especially thank Tim Robinson \(CSCS\) for access to and support of the computational resources\.

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