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

TL;DR

This paper analyzes the limitations of symmetry-adapted VQE methods for non-Abelian molecular point groups, revealing algebraic and numerical obstructions that cause systematic failures.

Contribution

It provides a Lie-algebraic explanation for the failure of symmetry-adapted VQE in non-Abelian cases and suggests necessary conditions for recovering full symmetry dynamics.

Findings

Symmetry-adapted VQE fails systematically for non-Abelian groups.

The Lie algebra restriction confines the reachable states to a measure-zero torus.

Numerical experiments confirm the algebraic restrictions and convergence issues.

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. For non-Abelian groups, they systematically fail, without a 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 $\mathfrak{u}(1)^{d_\lambda}$, restricting the reachable state manifold to a measure-zero torus $\mathbb{T}^{d_\lambda}$. However, completing the algebra is insufficient on its own, due to a numerical obstruction. Molecular orbitals adapted solely to an Abelian subgroup produce cross-component integrals that vanish identically, creating a zero-gradient plateau along non-Abelian algebraic directions. A proof-of-principle experiment on NH$_3$/STO-3G ($C_{3v}$, 16 qubits) confirms 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 an algebraic and geometric diagnosis of the observed numerical breakdown, establishing that recovering full equivariant dynamics requires both the inclusion of complete off-diagonal generators and the independent parametrization of cross-component excitations.