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Lower Bound on the Representation Complexity of Antisymmetric Tensor Product Functions

Yuyang Wang, Yukuan Hu, Xin Liu

TL;DR

The paper analyzes the representation complexity of antisymmetric tensor product functions (TPFs) in high dimensions, a central issue in quantum problems like the electronic Schrödinger equation. By linking antisymmetric TPFs to antisymmetric tensors and analyzing their CP ranks, it proves an exponential lower bound on the minimum number of TPF terms required to enforce antisymmetry, applicable to both discretized TPFs and tensor neural networks. The results explain why low-rank TPFs struggle to represent antisymmetric wave functions and justify determinant-based antisymmetric constructions; they quantify the bound as $\Theta(2^N/\sqrt{N})$ for nonzero antisymmetric functions. The findings guide future methods in high-dimensional quantum problems and motivate exploration of alternative tensor formats beyond low-rank TPFs.

Abstract

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.

Lower Bound on the Representation Complexity of Antisymmetric Tensor Product Functions

TL;DR

The paper analyzes the representation complexity of antisymmetric tensor product functions (TPFs) in high dimensions, a central issue in quantum problems like the electronic Schrödinger equation. By linking antisymmetric TPFs to antisymmetric tensors and analyzing their CP ranks, it proves an exponential lower bound on the minimum number of TPF terms required to enforce antisymmetry, applicable to both discretized TPFs and tensor neural networks. The results explain why low-rank TPFs struggle to represent antisymmetric wave functions and justify determinant-based antisymmetric constructions; they quantify the bound as for nonzero antisymmetric functions. The findings guide future methods in high-dimensional quantum problems and motivate exploration of alternative tensor formats beyond low-rank TPFs.

Abstract

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.
Paper Structure (11 sections, 10 theorems, 62 equations, 3 figures, 1 table)

This paper contains 11 sections, 10 theorems, 62 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tensor Product Function and Tensor Neural Network
  4. Antisymmetric Functions
  5. Antisymmetric Tensors
  6. Main results
  7. Representation Complexity of Antisymmetric Tensor Product Functions in Finite-Dimensional Spaces
  8. Representation Complexity of Antisymmetric Tensor Neural Networks
  9. Conclusion and Discussion
  10. Acknowledgements.
  11. Experimental Settings

Key Result

Proposition 2.1

Let $\mathbf{X} \in \mathcal{A}(\bigotimes^{N}\mathbb{C}^{K})$. Then:

Figures (3)

  • Figure 1: Numerical comparison of the TNN approximations with and without explicit antisymmetrization on the one-dimensional systems of $\mathrm{Li}$ (left) and $\mathrm{HeH}^+$ (right).
  • Figure 2: Architecture of TNN.
  • Figure 3: Comparison of TNN approximations with and without antisymmetrization for the one-dimensional Li system using $L=4$ and $m=40$, under the exponential decay learning rate schedule (Left) and the inverse time schedule (Right).

Theorems & Definitions (27)

  • Definition 2.1: TPF
  • Remark 2.1
  • Definition 2.2: TPF rank
  • Remark 2.2
  • Remark 2.3
  • Definition 2.3: CP Rank hitchcock1927expression
  • Proposition 2.1: hackbusch2012tensor
  • Remark 3.1
  • Lemma 3.1
  • proof
  • ...and 17 more