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 $16\times16$ square lattices.

• The paper presents a novel certification method that uses SDP relaxations to provide rigorous lower bounds on quantum spin ground-state energies.
• It leverages structured noncommutative polynomial optimization with symmetry exploitation and sparse monomial bases to reduce computational complexity.
• Numerical benchmarks show tight energy bounds on 1D chains and 16x16 lattices, offering practical certification for advanced quantum simulations.

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

Introduction and Context

The determination of ground-state properties in quantum many-body systems is a central issue in condensed matter physics, underpinning the understanding of low-temperature phases and quantum correlations. Although variational ansatz-based methods (DMRG, VMC, PEPS, NNQS) dominate current practice, their effectiveness is limited by ansatz expressivity. Exact diagonalization is unscalable, and QMC is hampered by the sign problem for frustrated/fermionic systems. Notably, variational techniques yield only upper bounds on ground-state energies, failing to guarantee the closeness to the true ground state or to provide certified error bounds for observables.

An alternative framework is provided by noncommutative polynomial optimization formalized in hierarchies of semidefinite programming (SDP) relaxations—most notably, the NPA hierarchy. Unlike variational approaches, SDP relaxations can yield rigorous lower bounds on ground states and certified intervals for observable expectation values. The dominant limitation of SDP-based certification has historically been scalability, as the size of the involved SDPs grows rapidly with system dimension and degree of relaxation.

This work presents an extensive advancement in tackling these limitations, combining noncommutative polynomial optimization with a thorough exploitation of the algebraic and physical symmetries of quantum spin systems. The result is a proven ability to certify ground-state properties on systems as large as $16 \times 16$ square lattices, greatly exceeding previous practical limits, and with notably tighter bounds.

Methodological Innovations

The paper details a hierarchy of SDP relaxations using sums of Hermitian squares (SOHS) to certify nonnegativity for the Hamiltonian minus a variable ground-state energy $\lambda$. For an $N$-site spin-$\frac{1}{2}$ system, all quantum operators are represented in the Pauli algebra and mapped to the corresponding matrix algebra via a canonical $\star$-isomorphism. The ground-state energy minimization is recast as a noncommutative polynomial optimization problem with commutation and anticommutation constraints from Pauli algebra. Each order of the relaxation ($d$) produces an SDP whose optimum $\lambda_d$ gives a systematically-improving lower bound on the ground-state energy.

Crucially, the approach leverages system structure to mitigate the combinatorial scaling inherent to SDP relaxations:

• Sparse Monomial Basis: The moment matrix is constructed only on monomials local in the Hamiltonian, discarding those with negligible or vanishing contributions due to sparsity. For 2D models, additional domain-specific monomials (e.g., small clusters) are included to retain relevant correlations.
• Symmetry Exploitation:
  • Sign Symmetries: Global and local sign flips in the spin operators yield block-diagonal structures in the moment matrices, reducing variable count by up to three orders of magnitude.
  • Permutation Symmetry: The invariance under $x\leftrightarrow y \leftrightarrow z$ exchange further consolidates equivalent SDP constraints, minimizing redundancy.
  • Translation and Lattice Symmetries: Periodic boundary conditions are exploited to reduce circulant blocks via Fourier analysis. In 2D, two rounds of block diagonalization exploit translation in both lattice directions.
  • Dihedral Symmetry: Operations in the symmetry group of the square lattice (rotations and reflections) are applied, further collapsing moment matrix dimension.
  • Conjugate Symmetry: The real (rather than complex) field is sufficient, enabling the use of more efficient real SDP solvers.
• Strengthening Constraints:
  • Reduced Density Matrix Positivity: Local reduced density matrices are enforced to be positive semidefinite, block-decomposed via $U(1)$ magnetization conservation.
  • State Optimality Conditions: Commutation relations and local stationarity (from the KKT conditions of state optimization) are imposed as additional linear or PSD constraints, as described in [araujo2023first, fawzi2024certified].

Numerical Results

The methodology is validated numerically across four standard Heisenberg-type models: 1D chain, 1D $J_1$-$\lambda$0 chain, 2D square lattice, and 2D $\lambda$1-$\lambda$2 model. Comparisons are made against state-of-the-art DMRG, QMC, and neural-network variational benchmarks.

Key outcomes include:

• Heisenberg Chain: For $\lambda$3 sites, the SDP lower bounds exhibit discrepancies below $\lambda$4 relative to DMRG upper bounds—orders of magnitude improvement over previous SDP bounds.
• 1D $\lambda$5-$\lambda$6 Chain: The gaps between SDP and DMRG energies shrink to below 0.01% in most regimes, with tight certified intervals for nearest- and next-nearest-neighbor correlations, and the structure factor.
• 2D Square Lattice: The method scales to $\lambda$7 lattices, previously unreachable by SDP-based approaches, with lower bounds systematically tighter than prior results for all $\lambda$8 and remaining within $\lambda$9 of the best known QMC estimates for $N$0.
• Observable Certification: Certified bounds on nonlocal correlators (e.g., at maximum Manhattan distance) systematically enclose the QMC/DMRG reference values, even when the observable falls outside the variational ansatz' direct reach.

**Importantly, the numerical results demonstrate that the exploitation of structure leads to several orders-of-magnitude reduction in maximal SDP block sizes (from 8 billion to 31 in the $N$1, $N$2 1D case), making problems accessible to standard hardware and solvers (MOSEK with 1T RAM).

Theoretical and Practical Implications

The present work establishes that rigorous, certified bounds for ground-state energies and observables are obtainable for quantum spin systems of sizes relevant to physically interesting regimes (2D lattices, $N$3). By leveraging symmetry, sparsity, and advanced constraint integration, SDP relaxations shed their former limitation to toy models. The framework provides a non-variational, complementary path to ground-state certification, immunized against false variational optima, and directly produces interval estimates for any polynomial observable.

The theoretical advances—such as full symmetry-adapted block-diagonalization and integration of the latest algebraic state optimality constraints—carry implications for future scalable quantum many-body certification methods.

On the practical side, this enables benchmarking of quantum simulation platforms and variational quantum algorithms, as well as cross-validation for DMRG and NNQS results in 1D and 2D. Notably, it creates the groundwork for certification in models where traditional methods break down (e.g., frustrated or sign-problematic systems).

Limitations and Directions for Future Work

While the structural exploitation is largely exhaustive, there remain avenues for enhancement:

• Incorporation of SU(2) Symmetry: As with U(1), utilizing the full non-Abelian symmetry could further collapse relaxation size and improve efficiency.
• Monomial Basis Selection: The choice of monomial basis crucially affects relaxation tightness for specific observables—systematic machine learning-guided selection (as discussed in [MLbounds]) deserves detailed investigation.
• Hybridization with Tensor-Network Methods: Integration of DMRG coarse-graining with SDP relaxations, following [kull2024lower], may extend certification to even larger systems.
• Application to Challenging Regimes: Testing on models and parameter regimes where sign-problems defeat QMC or variational ansatz are inexpressive, such as certain frustrated magnets or fermionic systems, is suggested as a high-impact direction.

Conclusion

The study demonstrates that noncommutative polynomial SDP relaxations, made scalable and tight via comprehensive exploitation of symmetry and algebraic structure, constitute a robust and practical methodology for certifying ground-state properties of quantum spin systems. The approach yields strong lower bounds and observable intervals rivaling leading numerical methods and opens the pathway to rigorous certification at lattice sizes and dimensions of physical interest. Continued methodological and algorithmic advances—particularly in basis selection, symmetry-adapted decomposition, and algorithmic integration with tensor-network methods—promise further expansion of the regime of certified quantum many-body physics (2604.01555).