A convergent hierarchy of spectral gap certificates for qubit Hamiltonians

arXiv:2510.08427v2 Announce Type: replace Abstract: 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 $\lambda_{0}(H) + \lambda_{1}(H)$. 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, Mass\'e, and Vol\v{c}i\v{c} 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.

Abstract: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 $\lambda_{0}(H) + \lambda_{1}(H)$. 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.

Quantum Physics (quant-ph); Data Structures and Algorithms (cs.DS)

arXiv:2510.08427 [quant-ph]

(or arXiv:2510.08427v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2510.08427 Focus to learn more arXiv-issued DOI via DataCite

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.

analiza aiqbench

A convergent hierarchy of spectral gap certificates for qubit Hamiltonians

Subiecte conexe