A convergent hierarchy of spectral gap certificates for qubit Hamiltonians

We give a convergent hierarchy of SDP certificates for bounding the spectral gap of local qubit Hamiltonians from below. 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 $\mathfrak{su}(2^{n})$, as well as additional constraints which put restrictions on the corresponding representations of the algebra. We also use as input an upper bound on the ground state energy, either using a hierarchy introduced by Fawzi, Fawzi, and Scalet, or an analog for qubit Hamiltonians of the Lasserre hierarchy of upper bounds introduced by Klep, Magron, Massé, and Volčič. The convergence of the certificates does not require that the Hamiltonian be frustration-free. We prove that the resulting certificates have polynomial size at fixed degree and converge asymptotically (in fact, at level $n$), by showing that all allowed representations of the algebra correspond to the second exterior power $\wedge^2(\mathbb{C}^{2^n})$, which encodes the sum of the two smallest eigenvalues of the original Hamiltonian. We also give an example showing that for a commuting 1-local Hamiltonian, the hierarchy certifies a nontrivial lower bound on the spectral gap.