This paper discusses the theoretical basis of variational algorithms for thinning degrees of freedom in a lattice Hamiltonian theory. We show that such a thinning can—in principle—be an exact real-space renormalization group transformation to a lattice with fewer sites. This exact transformation can only be constructed given the exact ground state of the theory. However, this insight teaches us how to significantly improve previous variational algorithms. Further improvements in ability to calculate quantities of interest is achieved by introducing "look-ahead" algorithms which allow us to maintain a renormallzationgroup interpretation while introducing long-distance physics into the determination of variational parameters.
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