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

Crosson, Elizabeth and Bowen, John (2017) Quantum ground state isoperimetric inequalities for the energy spectrum of local Hamiltonians. . (Submitted)

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We investigate the relationship between the energy spectrum of a local Hamiltonian and the geometric properties of its ground state. By generalizing a standard framework from the analysis of Markov chains to arbitrary (non-stoquastic) Hamiltonians we are naturally led to see that the spectral gap can always be upper bounded by an isoperimetric ratio that depends only on the ground state probability distribution and the range of the terms in the Hamiltonian, but not on any other details of the interaction couplings. This means that for a given probability distribution the inequality constrains the spectral gap of any local Hamiltonian with this distribution as its ground state probability distribution in some basis (Eldar and Harrow derived a similar result [1] in order to characterize the output of low-depth quantum circuits). Going further, we relate the Hilbert space localization properties of the ground state to higher energy eigenvalues by showing that the presence of k strongly localized ground state modes (i.e. clusters of probability, or subsets with small expansion) in Hilbert space implies the presence of k energy eigenvalues that are close to the ground state energy. Our results suggest that quantum adiabatic optimization using local Hamiltonians will inevitably encounter small spectral gaps when attempting to prepare ground states corresponding to multi-modal probability distributions with strongly localized modes, and this problem cannot necessarily be alleviated with the inclusion of non-stoquastic couplings.

Item Type:Report or Paper (Discussion Paper)
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Additional Information:E. C. is grateful for support provided by the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (NSF Grant PHY-1125565) with support of the Gordon and Betty Moore Foundation (GBMF-12500028). J.B. is grateful for support provided by the Caltech Summer Undergraduate Research Fellowship program, and also thanks the IQIM for hospitality.
Group:Institute for Quantum Information and Matter
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Gordon and Betty Moore FoundationGBMF-12500028
Record Number:CaltechAUTHORS:20171102-115811858
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:82896
Deposited By: Bonnie Leung
Deposited On:02 Nov 2017 19:13
Last Modified:02 Jun 2023 00:31

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