Bounds on the Ground State Energy of Quantum p-Spin Hamiltonians
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Author(s)
Anschuetz, Eric R.
Gamarnik, David
Kiani, Bobak T.
Date Issued
September 1, 2025
Journal
Communications in Mathematical Physics
Publisher
Springer Berlin Heidelberg
Citation
Anschuetz, E.R., Gamarnik, D. & Kiani, B.T. Bounds on the Ground State Energy of Quantum p-Spin Hamiltonians. Commun. Math. Phys. 406, 232 (2025).
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Author's final manuscript
Abstract
We consider the problem of estimating the ground state energy of quantum p-local spin glass random Hamiltonians, the quantum analogues of widely studied classical spin glass models. Our main result shows that the maximum energy achievable by product states has a well-defined limit (for even p) as n → ∞ and is E product ∗ = 2 log p in the limit of large p. This value is interpreted as the maximal energy of a much simpler so-called Random Energy Model, widely studied in the setting of classical spin glasses. The proof of the limit existing follows from an extension of Fekete’s Lemma after we demonstrate near super-additivity of the (normalized) quenched free energy. The proof of the value follows from a second moment method on the number of states achieving a given energy when restricting to an ϵ -net of product states. Furthermore, we relate the maximal energy achieved over all states to a p-dependent constant γ p , which is defined by the degree of violation of a certain asymptotic dependence ansatz over graph matchings. We show that the maximal energy achieved by all states E ∗ p in the limit of large n is at most γ p E product ∗ . We also prove using Lindeberg’s interpolation method that the limiting E ∗ p is robust with respect to the choice of the randomness and, for instance, also applies to the case of sparse random Hamiltonians. This robustness in the randomness extends to a wide range of random Hamiltonian models including SYK and random quantum max-cut.
MIT Department
Massachusetts Institute of Technology. Center for Theoretical Physics
Massachusetts Institute of Technology. Operations Research Center
Statistics and Data Science Center (Massachusetts Institute of Technology)
Sloan School of Management
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
https://doi.org/10.1007/s00220-025-05412-4
