Lower Bound on the Representation Complexity of Antisymmetric Tensor Product Functions
This provides a theoretical lower bound that explains the computational difficulty of using tensor product functions in quantum many-body problems, which require antisymmetry.
The paper proves that the minimum number of terms required for antisymmetric tensor product functions grows exponentially with problem dimension, showing that low-rank TPFs are fundamentally unsuitable for high-dimensional antisymmetric problems.
Tensor product function (TPF) approximations have been widely adopted in solving high-dimensional problems, such as partial differential equations and eigenvalue problems, achieving desirable accuracy with computational overhead that scales linearly with problem dimensions. However, recent studies have underscored the extraordinarily high computational cost of TPFs on quantum many-body problems, even for systems with as few as three particles. A key distinction in these problems is the antisymmetry requirement on the unknown functions. In the present work, we rigorously establish that the minimum number of involved terms for a class of TPFs to be exactly antisymmetric increases exponentially fast with the problem dimension. This class encompasses both traditionally discretized TPFs and the recent ones parameterized by neural networks. Our proof exploits the link between the antisymmetric TPFs in this class and the corresponding antisymmetric tensors and focuses on the Canonical Polyadic rank of the latter. As a result, our findings reveal that low-rank TPFs are fundamentally unsuitable for high-dimensional problems where antisymmetry is essential.