Complexity of Tensor Product Functions in Representing Antisymmetry (2501.05958v2)

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 uncover a fundamental incompatibility between antisymmetry and low-rank TPFs in high-dimensional contexts and offer new insights for further developments.

• The paper demonstrates that tensor product functions face an exponential complexity barrier in representing antisymmetry in finite-dimensional spaces.
• The authors link this complexity to the CP rank, proving that a minimal representation requires terms growing as O(2^N/√N).
• The study highlights significant implications for quantum mechanics, urging the pursuit of alternative models for high-dimensional antisymmetric systems.

Complexity of Tensor Product Functions in Representing Antisymmetry

The paper "Complexity of Tensor Product Functions in Representing Antisymmetry," authored by Yuyang Wang, Yukuan Hu, and Xin Liu, addresses the challenges posed by tensor product functions (TPFs) in capturing antisymmetry, specifically within quantum many-body problems. These problems are pivotal in quantum mechanics, given the necessity for antisymmetry in wave functions due to the Pauli exclusion principle.

Summary of Contributions

The authors present rigorous theoretical results demonstrating that TPFs face a substantial complexity barrier when representing antisymmetric functions. The key finding is that for the class of antisymmetric TPFs within finite-dimensional function spaces, the number of terms required grows exponentially with the dimension of the problem. This exponential dependency also extends to tensor neural networks (TNNs) engineered to construct such functions, emphasizing a fundamental limitation of TPFs in representing antisymmetry in high-dimensional settings.

Theoretical Insights

At the core of the paper is the establishment of a link between antisymmetric TPFs and their tensor representations, focusing on the Canonical Polyadic (CP) rank of these tensors. The CP rank provides a measure of the complexity of representing antisymmetric tensors, analogous to the TPF rank in function spaces.

The authors rigorously establish that for nonzero antisymmetric functions within any finite-dimensional space spanned by TPFs, the TPF rank, which denotes the minimal number of terms required for representation, is bounded below by an exponential function of the problem dimension:

• For a function $f$ that is antisymmetric and part of the space $\bigotimes^N F_K(\Omega)$, the TPF rank must satisfy $\text{rank}_{F_K(\Omega)}(f) \geq O(2^N/\sqrt{N})$.

This result applies even when TPFs are parameterized by neural networks (TNNs) with fixed architectures, implicating their inefficiency in representing antisymmetric properties accurately as dimensions increase.

Implications for Quantum Mechanics and Beyond

The implications of this research are significant for quantum mechanics, where constructing wave functions accurately under antisymmetry constraints is crucial. Given this limitation, traditional low-rank approximations or straightforward tensor-based models may be insufficient for high-dimensional quantum systems, such as those encountered in the electronic Schrödinger equation.

Practically, this suggests a necessity for alternative approaches beyond TPFs for modeling problems where antisymmetry is vital. This could involve leveraging determinant-based wave function constructions or hybrid models incorporating advanced tensor decompositions beyond conventional CP rank considerations.

Furthermore, understanding the complexity of representing antisymmetry can impact the development of new tensor formats or approximation strategies, optimally balancing computational efficiency and the fidelity of antisymmetry representation. It also raises questions about the potential trade-offs between computational resource allocation and approximation accuracy in practical scenarios.

Future Speculations

Looking forward, several avenues are intriguing for further exploration:

1. Enhancing Numerical Techniques: How can numerical techniques be adapted or extended to mitigate the exponential growth in complexity?
2. Exploring Alternative Formats: Could emerging tensor formats or representations offer a more compact or efficient means to handle antisymmetry in high-dimensional systems?
3. Quantifying Antisymmetry Errors: Developing methodologies to quantify and manage the error introduced when antisymmetry is approximated rather than exactly enforced could be crucial.
4. Neural Networks Adaptation: How might neural-based architectures evolve to accommodate the intrinsic limitations in expressing antisymmetry, potentially through novel architectures or hybrid approaches that integrate deterministic principles more robustly?

The paper's findings underscore a fundamental complexity inherent in TPFs and call for innovative advancements in both theoretical and computational methods in the ongoing effort to resolve high-dimensional quantum mechanics challenges robustly.