Extremal eigenvalues of local Hamiltonians

Aram W. Harrow1 and Ashley Montanaro2

1Center for Theoretical Physics, Department of Physics, MIT
2School of Mathematics, University of Bristol, UK

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We apply classical algorithms for approximately solving constraint satisfaction problems to find bounds on extremal eigenvalues of local Hamiltonians. We consider spin Hamiltonians for which we have an upper bound on the number of terms in which each spin participates, and find extensive bounds for the operator norm and ground-state energy of such Hamiltonians under this constraint. In each case the bound is achieved by a product state which can be found efficiently using a classical algorithm.

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[9] Cedric Yen-Yu Lin and Yechao Zhu, "Performance of QAOA on Typical Instances of Constraint Satisfaction Problems with Bounded Degree", arXiv:1601.01744, (2016).

[10] Thiago Bergamaschi, "Improved Product-state Approximation Algorithms for Quantum Local Hamiltonians", arXiv:2210.08680, (2022).

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