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Lie-algebraic incompleteness of symmetry-adapted VQE for non-Abelian molecular point groups

Leon D. da Silva, Marcelo P. Santos

Abstract

Symmetry-adapted variational quantum eigensolvers (VQE) based on the Unitary Coupled-Cluster ansatz (SymUCCSD) effectively reduce the parameter count for Abelian molecular point groups, yet they systematically fail for non-Abelian groups without a fully established theoretical explanation. In this work, we prove that the Abelian-subgroup restriction induces a spurious splitting of multidimensional irreducible representations, prematurely discarding cross-component excitations. At the Lie-algebraic level, this filter confines the Dynamical Lie Algebra (DLA) to the Abelian subalgebra $\mathfrak{u}(1)^{d_λ}$, restricting the reachable state manifold to a measure-zero torus $\mathbb{T}^{d_λ}$. However, completing the algebra is insufficient on its own due to a critical numerical trap: when standard molecular orbitals adapted solely to an Abelian subgroup are utilized, cross-component integrals vanish identically, creating an artificial zero-gradient plateau along non-Abelian algebraic directions. Numerical experiments on NH$_3$/STO-3G ($C_{3v}$, 16 qubits) confirm both the predicted DLA confinement and the gradient plateau, with SymUCCSD converging to an error of $21.8$~mHa above the FCI energy despite full optimizer convergence. Our analysis provides a rigorous algebraic and geometric diagnosis for the observed numerical breakdown, establishing that recovering full equivariant dynamics fundamentally necessitates both the inclusion of complete off-diagonal generators and the strategic parametrization of non-Abelian degrees of freedom.

Lie-algebraic incompleteness of symmetry-adapted VQE for non-Abelian molecular point groups

Abstract

Symmetry-adapted variational quantum eigensolvers (VQE) based on the Unitary Coupled-Cluster ansatz (SymUCCSD) effectively reduce the parameter count for Abelian molecular point groups, yet they systematically fail for non-Abelian groups without a fully established theoretical explanation. In this work, we prove that the Abelian-subgroup restriction induces a spurious splitting of multidimensional irreducible representations, prematurely discarding cross-component excitations. At the Lie-algebraic level, this filter confines the Dynamical Lie Algebra (DLA) to the Abelian subalgebra , restricting the reachable state manifold to a measure-zero torus . However, completing the algebra is insufficient on its own due to a critical numerical trap: when standard molecular orbitals adapted solely to an Abelian subgroup are utilized, cross-component integrals vanish identically, creating an artificial zero-gradient plateau along non-Abelian algebraic directions. Numerical experiments on NH/STO-3G (, 16 qubits) confirm both the predicted DLA confinement and the gradient plateau, with SymUCCSD converging to an error of ~mHa above the FCI energy despite full optimizer convergence. Our analysis provides a rigorous algebraic and geometric diagnosis for the observed numerical breakdown, establishing that recovering full equivariant dynamics fundamentally necessitates both the inclusion of complete off-diagonal generators and the strategic parametrization of non-Abelian degrees of freedom.
Paper Structure (13 sections, 7 theorems, 9 equations, 1 figure, 1 table)

This paper contains 13 sections, 7 theorems, 9 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Algebraic setup and definitions
  3. The Abelian case: correctness of SymUCCSD
  4. The non-Abelian case: linear incompleteness
  5. Topological consequence: confinement of the DLA
  6. Explicit illustration: inaccessibility in NH$_3$ ($C_{3v}$)
  7. The Abelian Basis Trap and Requirements for a Complete Ansatz
  8. Numerical validation: NH$_3$/STO-3G
  9. Discussion
  10. Relationship to HiUCCSD
  11. Connection to equivariant quantum circuits
  12. Limitations and outlook
  13. Conclusion

Key Result

Theorem 3.1

Let $G$ be an Abelian point group. Then $\mathcal{P}_{\mathrm{Sym}} = \mathcal{P}_{G}$, and every anti-Hermitian $G$-equivariant operator in the UCCSD space can be written as a linear combination of operators in $\mathcal{P}_{\mathrm{Sym}}$.

Figures (1)

  • Figure 1: Conceptual graphical abstract diagnosing the gradient trap. (Note on notation: In the schematic, $[H,G]$ refers to the commutator between the Hamiltonian $\mathcal{H}$ and the unitary Lie generator $\hat{G}_k$, and should not be confused with the molecular point groups $H$ and $G$). In SymUCCSD (Left), an Abelian-adapted MO basis trivializes the gradient commutator $\bra{\mathrm{HF}} [\mathcal{H}, \hat{G}_k] \ket{\mathrm{HF}} = 0$, forcing the optimizer into a flat energy plateau. Conversely, establishing a complete $G$-equivariant ansatz alongside rotated MOs or fully independent parameterization restores the non-zero gradients, allowing successful convergence to the global correlation minimum (Right).

Theorems & Definitions (22)

  • Definition 2.1: Group action on Fock space
  • Definition 2.2: Orbital partitioning
  • Definition 2.3: Hamiltonian and integrals
  • Definition 2.4: UCCSD pool and adjoint action
  • Definition 2.5: Symmetry filters: $G$-equivariant vs. SymUCCSD
  • Definition 2.6: Trotterized ansatz and Dynamical Lie Algebra
  • Theorem 3.1: Equivalence for Abelian groups
  • proof
  • Remark 3.2
  • Lemma 4.1: Irrep splitting under Abelian restriction
  • ...and 12 more