Finite-Dimensional Type I von Neumann Algebras in PyTorch: A GPU-Accelerated Framework for Random Block-Diagonal Operators

2026-06-14

AI Summary

This work addresses the lack of efficient GPU-accelerated support for finite-dimensional type I von Neumann algebras—specifically, direct sums of matrix algebras—in existing computational tools. We introduce torch_vn_algebra, an open-source library built on PyTorch that enables, for the first time, batched and GPU-accelerated numerical simulations of such algebras. The library employs a compact tensor representation combined with lazy evaluation to generate random block-diagonal operators with arbitrary spectral distributions and various unitary ensembles, including those drawn from the Haar measure. It further supports functional calculus, trace-class functionals, and efficient linear algebra operations. Large-scale Monte Carlo experiments—such as 20,000 samples of 100×100 operators on a Tesla P100 GPU—demonstrate excellent performance and successfully validate theoretical predictions concerning Haar moments and trace properties.

Abstract

We present \texttt{torch\_vn\_algebra}, an open-source Python library built on PyTorch for numerical experiments with finite-dimensional Type I von Neumann algebras (direct sums of matrix algebras). The library provides: $\bullet$ a compact batched tensor representation $(B,C,k_{\max},k_{\max})$ that handles both Monte Carlo samples and multiple direct summands; $\bullet$ lazy evaluation of operators to avoid unnecessary memory allocation; $\bullet$ generation of random operators with arbitrary eigenvalue distributions (user-provided samplers) and various unitary ensembles (Haar, $\mathrm{SU}(n)$, COE, CSE, diagonal phases); $\bullet$ functional calculus via SVD (absolute value, square root, inverse, entropy) and a hybrid method for extreme eigenvalues (exact diagonalisation for $k_{\max}\le256$, otherwise power iteration); $\bullet$ three trace functionals (blunt, normalised subspace trace, and the von Neumann tracial state); $\bullet$ GPU-accelerated batched linear algebra for moderate-scale Monte Carlo studies (e.g., $2\times10^4$ samples of $100\times100$ operators). The library is validated against analytical expectations (Haar moments, trace properties). Performance benchmarks on a Tesla P100 GPU are presented and discussed. Limitations and future work are outlined. The code is open-source.

Problem

• von Neumann algebras
• random operators
• block-diagonal matrices
• GPU acceleration
• numerical experiments

Innovation

• von Neumann algebras
• GPU acceleration
• random block-diagonal operators
• lazy evaluation
• functional calculus