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Scalable Ground-State Certification of Quantum Spin Systems via Structured Noncommutative Polynomial Optimization

Jie Wang, David Jansen, Irénée Frerot, Marc-Olivier Renou, Victor Magron, Antonio Acín

Abstract

A fundamental challenge in quantum physics is determining the ground-state properties of many-body systems. Whereas standard approaches, such as variational calculations, consist of writing down a wave function ansatz and minimizing over the possible states expressible by this ansatz, one can alternatively formulate the problem as a noncommutative polynomial optimization problem. This optimization problem can then be addressed using a hierarchy of semidefinite programming relaxations. In contrast to variational calculations, the semidefinite program can provide lower bounds for ground state energies and upper and lower bounds on observable expectation values. However, this approach typically suffers from severe scalability issues, limiting its applicability to small-to-medium-scale systems. In this article, we demonstrate that leveraging the inherent structures of the system can significantly mitigate these scalability challenges and thus allows us to compute meaningful bounds for quantum spin systems on up to $16\times16$ square lattices.

Scalable Ground-State Certification of Quantum Spin Systems via Structured Noncommutative Polynomial Optimization

Abstract

A fundamental challenge in quantum physics is determining the ground-state properties of many-body systems. Whereas standard approaches, such as variational calculations, consist of writing down a wave function ansatz and minimizing over the possible states expressible by this ansatz, one can alternatively formulate the problem as a noncommutative polynomial optimization problem. This optimization problem can then be addressed using a hierarchy of semidefinite programming relaxations. In contrast to variational calculations, the semidefinite program can provide lower bounds for ground state energies and upper and lower bounds on observable expectation values. However, this approach typically suffers from severe scalability issues, limiting its applicability to small-to-medium-scale systems. In this article, we demonstrate that leveraging the inherent structures of the system can significantly mitigate these scalability challenges and thus allows us to compute meaningful bounds for quantum spin systems on up to square lattices.

Paper Structure

This paper contains 21 sections, 14 theorems, 68 equations, 9 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Noncommutative polynomials
  4. The Pauli algebra for quantum $\frac{1}{2}$-spin systems
  5. Ground-state certification via noncommutative polynomial optimization
  6. Reducing SDP sizes by exploiting structures
  7. Sparsity
  8. Symmetry
  9. Sign symmetry of the model
  10. Sign symmetry of the Hamiltonian
  11. Conjugate symmetry of the Hamiltonian
  12. Translation symmetry of the lattices
  13. Permutation symmetry on $x,y,z$
  14. Mirror/dihedral symmetry of the lattices
  15. Strengthening the SDP relaxations
  16. ...and 6 more sections

Key Result

Lemma 2.1

Let $f\in{\mathbb{C}}\langle{\mathbf{x}}\rangle$ with $\deg(f)=2d$. Then $f$ is a SOHS if and only if there exists $G\in{\mathbf{H}}_+^{|W_d|}$ ($|W_d|$ stands for the dimension of $W_d$) satisfying where $W_d^{\star}$ is the row vector consisting of $u^{\star}, u\in W_d$. $\blacktriangleleft$$\blacktriangleleft$

Figures (9)

  • Figure 1: Lattice sites of three-body monomials appearing in the monomial basis for the 2D case.
  • Figure 1: Bounds on ground-state energies for the Heisenberg chain with $N$ spins.
  • Figure 2: Bounds on ground-state energies for the $J_1$-$J_2$ Heisenberg chain ($N=40$).
  • Figure 3: Bounds on $C(1)$ for the $J_1$-$J_2$ Heisenberg chain ($N=40$).
  • Figure 4: Bounds on $C(2)$ for the $J_1$-$J_2$ Heisenberg chain ($N=40$).
  • ...and 4 more figures

Theorems & Definitions (28)

  • Lemma 2.1: helton2002positive, Lemma 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Proof 1
  • Remark 3.1
  • Proposition 4.1
  • Proof 2
  • Lemma 4.2
  • Proof 3
  • Lemma 4.3
  • ...and 18 more