We give a convergent hierarchy of SDP certificates for lower bounds on the spectral gap of a local qubit Hamiltonian $H$. Our approach is based on the NPA hierarchy applied to a polynomially-sized system of constraints defining the universal enveloping algebra of the Lie algebra $\mathfrak{su}(2^{n})$ along with additional constraints which force the representation to be $\wedge^{2}(\mathbb{C}^{2^{n}})$, one which the induced Hamiltonian has ground energy $λ_{0}(H) + λ_{1}(H)$. Combined with an upper bound on the ground state energy, which can be obtained either using a hierarchy introduced by Fawzi, Fawzi, and Scalet or a noncommutative analog developed by Klep, Magron, Massé, and Volčič of a hierarchy introduced by Lasserre, we obtain lower bounds on the spectral gap of $H$ without assuming frustration-freeness or geometric locality. We prove that the resulting certificates have polynomial size at fixed degree and converge at level $n$. As illustrations of the certificates which can be obtained, we show that for a commuting 1-local Hamiltonian the hierarchy certifies the optimal bound on its spectral gap and prove that similar hierarchies require degree growing with $n$ to certify the same bound. We then extend this to prove a lower bound on the spectral gap of the transverse-field $XY$ model, which is not frustration-free, on arbitrary regular graphs without any assumption of geometric locality.

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