A convergent hierarchy of spectral gap certificates for qubit Hamiltonians

• URL: http://arxiv.org/abs/2510.08427v1
• Date: Thu, 09 Oct 2025 16:42:21 GMT
• Title: A convergent hierarchy of spectral gap certificates for qubit Hamiltonians
• Authors: Sujit Rao,
• Abstract summary: We give a hierarchy of SDP certificates for bounding the spectral gap of local qubit Hamiltonians from below.<n>We prove that the resulting certificates have size at fixed degree and convergeally (in fact, at level $n$)<n>We also give an example showing that for a commuting 1-local Hamiltonian, the hierarchy certifies a non lower bound on the spectral gap.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-sa/4.0/
• Abstract: 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, Mass\'{e}, and Vol\v{c}i\v{c}. 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.

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