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We investigate the relationship between the energy spectrum of a localHamiltonian and the geometric properties of its ground state. By generalizing astandard framework from the analysis of Markov chains to arbitrary(non-stoquastic) Hamiltonians we are naturally led to see that the spectral gapcan always be upper bounded by an isoperimetric ratio that depends only on theground state probability distribution and the range of the terms in theHamiltonian, but not on any other details of the interaction couplings. Thismeans that for a given probability distribution the inequality constrains thespectral gap of any local Hamiltonian with this distribution as its groundstate probability distribution in some basis (Eldar and Harrow derived asimilar result in order to characterize the output of low-depth quantumcircuits). Going further, we relate the Hilbert space localization propertiesof the ground state to higher energy eigenvalues by showing that the presenceof k strongly localized ground state modes (i.e. clusters of probability, orsubsets with small expansion) in Hilbert space implies the presence of k energyeigenvalues that are close to the ground state energy. Our results suggest thatquantum adiabatic optimization using local Hamiltonians will inevitablyencounter small spectral gaps when attempting to prepare ground statescorresponding to multi-modal probability distributions with strongly localizedmodes, and this problem cannot necessarily be alleviated with the inclusion ofnon-stoquastic couplings.

Quantum ground state isoperimetric inequalities for the energy spectrum of local Hamiltonians