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Numerical Algebraic Geometry for Energy Computations on Tensor Train Varieties

Viktoriia Borovik, Hannah Friedman, Serkan Hoşten, Max Pfeffer

TL;DR

The paper develops a rigorous algebraic-geometry framework for energy computations in quantum chemistry by minimizing the Rayleigh quotient $R_H(\psi)$ over tensor train varieties $V_{\mathbf{k},\mathbf{r}}$, introducing the RR degree as the count of critical points for generic $H$ and the RR discriminant describing degenerate cases. It provides a complete geometric picture: a birational parametrization of TT manifolds from Grassmannians, a parametric and implicit approach to critical point computation, and the RR correspondence and Bombieri-Weyl viewpoint relating critical points to discriminants and distances in Veronese space. Numerical experiments using homotopy continuation quantify RR degrees for small TT and determinantal varieties, reveal high discriminant degrees, and benchmark ALS/DMRG; they show ALS often hits local minima and DMRG may fail to converge to TT-manifold critical points. The results offer theoretical guarantees and practical guidance for TT-based energy computations, highlighting when and how critical-point structure governs the success of common tensor-network optimization techniques in quantum chemistry.

Abstract

We study energy minimization problems in quantum chemistry through the lens of computational algebraic geometry. We focus on minimizing the Rayleigh quotient of a Hamiltonian over a tensor train variety. The complex critical points of this problem approximate eigenstates of the quantum system, with the global minimum approximating the ground state. We call the number of critical points the Rayleigh-Ritz degree. After introducing tensor train varieties, we identify instances when they are Segre products of projective spaces. We also report what we know about the defining ideals of tensor trains. We present a birational parametrization of them from products of Grassmannians. Along the way, we study the Rayleigh-Ritz degree, and we introduce the Rayleigh-Ritz discriminant, which describes Hamiltonians that lead to deficient number of critical points. We use homotopy continuation to compute all critical points of this optimization problem over various tensor train and determinantal varieties. Finally, we use these results to benchmark state-of-the-art methods, the Alternating Linear Scheme and Density Matrix Renormalization Group.

Numerical Algebraic Geometry for Energy Computations on Tensor Train Varieties

TL;DR

The paper develops a rigorous algebraic-geometry framework for energy computations in quantum chemistry by minimizing the Rayleigh quotient over tensor train varieties , introducing the RR degree as the count of critical points for generic and the RR discriminant describing degenerate cases. It provides a complete geometric picture: a birational parametrization of TT manifolds from Grassmannians, a parametric and implicit approach to critical point computation, and the RR correspondence and Bombieri-Weyl viewpoint relating critical points to discriminants and distances in Veronese space. Numerical experiments using homotopy continuation quantify RR degrees for small TT and determinantal varieties, reveal high discriminant degrees, and benchmark ALS/DMRG; they show ALS often hits local minima and DMRG may fail to converge to TT-manifold critical points. The results offer theoretical guarantees and practical guidance for TT-based energy computations, highlighting when and how critical-point structure governs the success of common tensor-network optimization techniques in quantum chemistry.

Abstract

We study energy minimization problems in quantum chemistry through the lens of computational algebraic geometry. We focus on minimizing the Rayleigh quotient of a Hamiltonian over a tensor train variety. The complex critical points of this problem approximate eigenstates of the quantum system, with the global minimum approximating the ground state. We call the number of critical points the Rayleigh-Ritz degree. After introducing tensor train varieties, we identify instances when they are Segre products of projective spaces. We also report what we know about the defining ideals of tensor trains. We present a birational parametrization of them from products of Grassmannians. Along the way, we study the Rayleigh-Ritz degree, and we introduce the Rayleigh-Ritz discriminant, which describes Hamiltonians that lead to deficient number of critical points. We use homotopy continuation to compute all critical points of this optimization problem over various tensor train and determinantal varieties. Finally, we use these results to benchmark state-of-the-art methods, the Alternating Linear Scheme and Density Matrix Renormalization Group.

Paper Structure

This paper contains 14 sections, 32 theorems, 89 equations, 1 figure, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Geometry of Tensor Train Varieties
  3. Birational Parametrization and TT manifolds
  4. Rayleigh-Ritz Optimization and the RR degree
  5. The Lagrangian Locus
  6. Parametric Critical Locus
  7. Rayleigh-Ritz Correspondence and Discriminant
  8. Rayleigh-Ritz Correspondence
  9. Rayleigh-Ritz Discriminant
  10. Bombieri-Weyl Correspondence
  11. Numerical Experiments
  12. Hamiltonians
  13. Homotopy Continuation Computations
  14. DMRG and ALS

Key Result

Lemma 2.2

The tensor train variety $V_{\mathbf{k}, \mathbf{r}}$ is equal to In particular, set-theoretically $V_{\mathbf{k}, \mathbf{r}}$ is cut by all $(r_{i+1} + 1)\textrm{-minors of flattenings $\psi^i$}$.

Figures (1)

  • Figure 1: A tensor network diagram for binary tensors of order 4 of TT rank $(1,2,1)$.

Theorems & Definitions (95)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Example 2.4
  • Proposition 2.5
  • proof
  • Theorem 2.6
  • proof
  • ...and 85 more