Exponential scaling of the unconstrained MIS-QAOA dynamical Lie algebra

Determine whether the dynamical Lie algebra generated by the unconstrained edge-penalty Maximum Independent Set problem Hamiltonian and the transverse-field mixer has exponential dimension in the number of vertices for generic connected graphs.

Background

The paper discusses prior work on unconstrained QAOA for Maximum Independent Set using an edge-penalty Hamiltonian and a transverse-field mixer. It states that the associated dynamical Lie algebra contains the MaxCut dynamical Lie algebra as a subalgebra, and that exponential dimension has been conjectured for generic connected graphs. This remains a conjectural scaling claim rather than an established theorem in the cited discussion.

References

The resulting dynamical Lie algebra $\mathfrak{g}$ contains the MaxCut DLA as a subalgebra (which follows directly from the proof technique of Lemma~A.5) and is thus widely conjectured to scale exponentially in dimension with $n$ for generic connected graphs (see).

— The Expressive Power of Constrained QAOA: What You Might Have MISsed  (2609.18209 - Tsvelikhovskiy et al., 16 Sep 2026) in Section 1, Technical overview