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.
 N. Bansal, S. Bravyi, and B. Terhal. Classical approximation schemes for the ground-state energy of quantum and classical Ising spin hamiltonians on planar graphs. Quantum Inf. Comput., 9(8):0701-0720, 2009, arXiv:0705.1115.
 B. Barak, A. Moitra, R. O'Donnell, P. Raghavendra, O. Regev, D. Steurer, L. Trevisan, A. Vijayaraghavan, D. Witmer, and J. Wright. Beating the random assignment on constraint satisfaction problems of bounded degree, 2015, arXiv:1505.03424.
 F. G. S. L. Brandão and A. W. Harrow. Product-state approximations to quantum ground states. Communications in Mathematical Physics, 342(1):47-80, 2006. arXiv:1310.0017.
 I. Dinur, E. Friedgut, G. Kindler, and R. O'Donnell. On the Fourier tails of bounded functions over the discrete cube. Israel Journal of Mathematics, 160:389-412, 2007.
 E. Farhi, J. Goldstone, and S. Gutmann. A quantum approximate optimization algorithm applied to a bounded occurrence constraint problem. Technical Report MIT-CTP/4628, MIT, 2014, arXiv:1412.6062.
 M. Hartmann, G. Mahler, and O. Hess. Gaussian quantum fluctuations in interacting many particle systems. Letters in Mathematical Physics, 68(2):103-112, 2004, arXiv:math-ph/0312045.
 A. Montanaro. Some applications of hypercontractive inequalities in quantum information theory. Journal of Mathematical Physics, 53(12):122206, 2012, arXiv:1208.0161.
 R. O'Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014.
 J. Renes, R. Blume-Kohout, A. J. Scott, and C. Caves. Symmetric informationally complete quantum measurements. J. Math. Phys., 45(6):2171-2180, 2003. quant-ph/0310075.
 Sergey Bravyi, David Gosset, Robert König, and Kristan Temme, "Approximation algorithms for quantum many-body problems", Journal of Mathematical Physics 60 3, 032203 (2019).
 Cedric Yen-Yu Lin and Yechao Zhu, "Performance of QAOA on Typical Instances of Constraint Satisfaction Problems with Bounded Degree", arXiv:1601.01744.
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