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Given a renormalization scheme, we show how to formulate a tractable convexrelaxation of the set of feasible local density matrices of a many-body quantumsystem. The relaxation is obtained by introducing a hierarchy of constraintsbetween the reduced states of ever-growing sets of lattice sites. Thecoarse-graining maps of the underlying renormalization procedure serve toeliminate a vast number of those constraints, such that the remaining ones canbe enforced with reasonable computational means. This can be used to obtainrigorous lower bounds on the ground state energy of arbitrary localHamiltonians, by performing a linear optimization over the resulting convexrelaxation of reduced quantum states. The quality of the bounds cruciallydepends on the particular renormalization scheme, which must be tailored to thetarget Hamiltonian. We apply our method to 1D translation-invariant spinmodels, obtaining energy bounds comparable to those attained by optimizing overlocally translation-invariant states of $n\gtrsim 100$ spins. Beyond thisdemonstration, the general method can be applied to a wide range of otherproblems, such as spin systems in higher spatial dimensions, electronicstructure problems, and various other many-body optimization problems, such asentanglement and nonlocality detection.

Lower Bounding Ground-State Energies of Local Hamiltonians Through the Renormalization Group