Reductions of QAOA Induced by Classical Symmetries: Theoretical Insights and Practical Implications

The performance of the Quantum Approximate Optimization Algorithm (QAOA) is closely tied to the structure of the dynamical Lie algebra (DLA) generated by its Hamiltonians, which determines both its expressivity and trainability. In this work, we show that classical symmetries can be systematically exploited as a design principle for QAOA. Focusing on the MaxCut problem with global bit-flip symmetry, we analyze reduced QAOA instances obtained by fixing a single variable and study how this choice affects the associated DLAs. We show that the structure of the DLAs can change dramatically depending on which variable is held fixed. In particular, we construct explicit examples where the dimension collapses from exponential to quadratic, uncovering phenomena that do not appear in the original formulation. Numerical experiments on asymmetric graphs indicate that such reductions often produce DLAs of much smaller dimension, suggesting improved trainability. We also prove that any graph can be embedded into a slightly larger one (requiring only quadratic overhead) such that the standard reduced DLA coincides with the free reduced DLA, in most cases implying exponential dimension and irreducibility on the Hilbert space for reduced QAOA instances. These results establish symmetry-aware reduction as a principled tool for designing expressive and potentially trainable QAOA circuits.