Tensor

A tensor is a multidimensional array that generalizes scalars (0D), vectors (1D), and matrices (2D) to higher dimensions. Tensors are widely used in fields like physics, machine learning, and computational chemistry because they allow the efficient representation and manipulation of data that naturally occurs in multiple dimensions.

In many-body perturbation theory (MBPT), dealing with high-dimensional tensors (such as electron integrals) becomes computationally expensive due to the large amount of memory and computational power required. Tensor decompositions offer a way to reduce this complexity by approximating high-dimensional tensors with lower-dimensional representations, while maintaining accuracy.

One popular decomposition method is the Tensor Train (TT) decomposition, where a high-dimensional tensor is factored into a sequence of low-rank matrices, known as “cores.” This significantly reduces storage and computational costs. TT decomposition allows tensors to be manipulated efficiently and is particularly useful in MBPT, where electron integrals and correlation functions are naturally tensorial in structure.

Other decomposition methods, such as canonical polyadic (CP) decomposition and hierarchical Tucker (HT) decomposition, similarly approximate tensors by breaking them into simpler components. These methods can dramatically reduce the cost of tensor operations and enable faster computations in MBPT by efficiently handling the large-scale tensors involved in electron interactions.

By leveraging tensor decompositions, MBPT calculations can be made scalable and more computationally feasible, particularly when working with systems involving complex interactions.