Towards a universal gateset for mathsf{QMA}₁

mathsf{QMA}₁ is mathsf{QMA} with perfect completeness, i.e., the prover must accept with a probability of exactly 1 in the YES-case. Whether mathsf{QMA}₁ and mathsf{QMA} are equal is still a major open problem. It is not even known whether mathsf{QMA}₁ has a universal gateset; Solovay-Kitaev does not apply due to perfect completeness. Hence, we do not generally know whether mathsf{QMA}₁^G=mathsf{QMA}₁^G' (superscript denoting gateset), given two universal gatesets G,G'. In this paper, we make progress towards the gateset question by proving that for all kinmathbb N, the gateset G_2^k (Amy et al., RC 2024) is universal for all gatesets in the cyclotomic field mathbb{Q}\(ζ_2^k\),ζ_2^k=e^2πi/2k, i.e. mathsf{QMA}₁^{Gsubseteqmathsf}{QMA}₁^G_2k for all gatesets G in mathbb{Q}\(ζ_2^k\). For mathsf{BQP}₁, we can even show that G₂ suffices for all 2^k-th cyclotomic fields. We exhibit complete problems for all mathsf{QMA}₁^G_2k: Quantum l-SAT in mathbb{Q}\(ζ_2^k\) is complete for mathsf{QMA}₁^G_2k for all lge4, and l=3 if kge3, where quantum l-SAT is the problem of deciding whether a set of l-local Hamiltonians has a common ground state. Additionally, we give the first mathsf{QMA}₁-complete 2-local Hamiltonian problem: It is mathsf{QMA}₁^G_2k-complete for $kge3$ to decide whether a given 2-local Hamiltonian H in mathbb{Q}\(ζ_2^k\) has a nonempty nullspace. Our techniques also extend to sparse Hamiltonians, and so we can prove the first mathsf{QMA}₁(2)-complete i.e. $mathsf{QMA}₁$ with two unentangled provers Hamiltonian problem. Finally, we prove that the Gapped Clique Homology problem defined by King and Kohler (FOCS 2024) is mathsf{QMA}₁^G₂-complete, and the Clique Homology problem without promise gap is PSPACE-complete.