Standardised Tensor Calculations Promise Faster Simulations for Materials and Physics Research

Researchers are increasingly reliant on tensor computations across diverse scientific domains, necessitating standardised approaches to facilitate efficient and portable code. Jan Brandejs (CNRS, University of Toulouse, France), Trond Saue (CNRS, University of Toulouse, France) and Andre Severo Pereira Gomes (Universit e de Lille, CNRS, UMR 8523, PhLAM, Physique des Lasers, Atomes et Mol ecules, F-59000 Lille, France) et al. recently convened the second Toulouse Tensor Workshop to address this growing need. Building upon the momentum generated by the first workshop and the subsequent formation of a tensor standardisation working group, this meeting, held on 17-19 September 2025, provided a crucial forum for application developers and software experts to refine a proposed low-level interface for tensor operations. The discussions and feedback gathered represent a significant step towards establishing a robust and widely adopted standard, ultimately accelerating progress in fields such as materials science and electronic structure calculations.

This achievement addresses a growing need for interoperability and performance in tensor algebra, which underpins complex calculations in areas like materials science, electronic structure, and quantum many-body simulations.

The work culminated in the formation of the Tensor Algebra Processing Primitives Working Group, or TAPP-WG, bringing together experts from both academic institutions and industry to collaboratively define and implement this new standard. Following the initial workshop in May 2024, the TAPP-WG meticulously specified a low-level interface, making it freely available on GitHub for broad community access and contribution.
This interface allows for streamlined communication between different tensor libraries and applications, reducing the overhead associated with data conversion and enabling more efficient use of computational resources. The subsequent workshop, held in September 2025, served as a platform for gathering feedback on this interface and charting a course for future development, including higher-level functionalities and support for specialised tensor formats.

The establishment of the TAPP-WG, consisting of experts from academia and industry, represents a key number in this endeavour, signifying a dedicated and collaborative approach to standardisation. This collaborative effort ensures the interface meets the diverse needs of the scientific community and promotes long-term sustainability.

The reference implementation, detailed in a recently published whitepaper, provides a concrete foundation for developers to build upon and integrate into their existing workflows. This standardised interface promises to accelerate scientific discovery by simplifying the development of high-performance tensor-based applications.

By removing barriers to interoperability, researchers can focus on innovation rather than wrestling with compatibility issues. The work also paves the way for tackling more complex computational challenges, such as those involving sparse tensors and advanced decomposition techniques, ultimately enabling breakthroughs in fields reliant on intensive numerical computation.

Establishing the Tensor Application Programming Interface Working Group and surveying tensor contraction requirements

Researchers initiated a program to standardise low-level interfaces for tensor operations, culminating in the establishment of the Tensor Application Programming Interface Working Group, or TAPP-WG, following a workshop in 2024. This group, comprised of experts from both academic institutions and industrial partners, was formed to specifically develop this crucial standard interface.

The methodology centred on a two-pronged approach: the creation of the TAPP-WG and the organisation of a detailed survey on tensor contractions. This collaborative effort aimed to understand the needs of both developers and users of tensor software, ultimately driving the design of a robust and widely applicable standard.

The TAPP-WG, consisting of Jan Brandejs, Devin Matthews, Edward Valeev, Paul Springer, Paolo Bientinesi, Justin Turney, Alexander Heinecke, Christopher Milette, Lucas Visscher, Jeff Hammond, and Niklas Hornblad, meticulously established a low-level interface, documented on a dedicated website hosting discussions and the Reference TAPP implementation. A detailed whitepaper was produced, providing comprehensive information on TAPP and its reference implementation.

This interface was then integrated into a real-world application, the ExaCorr coupled cluster module of the DIRAC electronic structure code, through the definition of Fortran bindings to the C API and a higher-level interface utilising einsum notation. Concurrently, a survey was distributed to 130 contacts, garnering 80 responses, to map the landscape of tensor software users and developers.

The survey revealed that 69% of respondents were either developers, users, or both, and that tensor contractions significantly impacted application performance in most cases. Crucially, 69% of respondents indicated that tensors with at least three indices were central to performance-critical code sections.

The data, accessible via Zenodo, highlighted a preference for Einstein-based notation (43%) for high-level interfaces and widespread use of Python/Julia libraries like NumPy and PyTorch. This follow-up meeting focused on refining a low-level standard interface for tensor operations, a crucial step towards streamlining tensor computations across diverse scientific domains.

The establishment of the TAPP-WG, a working group comprising experts from both academia and industry, represents a key outcome of this collaborative effort, with 1 working group formed following the first workshop in 2024 to specifically develop this low-level standard interface. A comprehensive survey conducted as part of the workshop revealed that 69% of respondents indicated tensor contractions strongly influence the performance of their applications.

Furthermore, the survey data showed that 69% of those surveyed are either developers of application software, users of these, or both, highlighting the broad relevance of efficient tensor operations. The study also determined that tensors with at least three indices typically represent the performance-critical parts of the code for most respondents.

Analysis of the survey responses indicated that 63% of respondents currently utilize automatic differentiation techniques in their work. The data further revealed that 66% of use cases involve a small number of very large tensors, while 27% involve a large number, exceeding 1000, of smaller tensors. Regarding data structure, 34% of respondents work with dense tensors, and 43% exploit block sparsity in their computations.

These findings suggest a community need for improvements in handling tensor contractions on single nodes and enhancing the exploitation of block sparsity and index permutational symmetries. The potential adoption of the TAPP interface as a backend for widely used frameworks like NumPy and PyTorch was also discussed as a means of broadening its impact and accessibility. The implementation of TAPP-torch, a PyTorch operation extension for TAPP, represents an initial step in this direction.

TAPP standardisation enables collaborative tensor library development and application integration

The establishment of a working group, the TAPP-WG, consisting of experts from academia and industry, formed following the first workshop in 2024, represents a significant step towards standardising tensor operations. This group has specified a low-level standard interface for these operations, freely available on GitHub, facilitating broader collaboration and code reusability.

The recent workshop in Toulouse further refined this interface based on feedback from developers of applications reliant on tensor computations, such as those used in materials science and electronic structure calculations. This standardised interface is highly valuable to the scientific community as it provides a foundation upon which additional functionality can be built.

A reference implementation of the TAPP API in the C programming language currently supports several tensor libraries as backends and has been successfully integrated into the DIRAC relativistic molecular electronic structure code with minimal modification. Participants acknowledged the need to incorporate support for block sparsity as the next crucial step in the standardisation process, while considering efforts to address more general sparsity types and distributed memory calculations premature at this stage.

The authors recognise limitations in the current scope of TAPP, specifically regarding support for complex sparsity patterns and distributed memory computations. Future research will focus on incorporating block sparsity and exploring the feasibility of an einsum-based format for user interaction. Additionally, the potential for integrating automatic differentiation capabilities is being considered, potentially through the formation of a dedicated workgroup to address this feature.