  We consider a class of holographic tensor networks that are efficiently
contractible variational ansatze, manifestly (approximate) quantum error
correction codes, and can support power-law correlation functions. In the case
when the network consists of a single type of tensor that also acts as an
erasure correction code, we show that it cannot be both locally contractible
and sustain power-law correlation functions. Motivated by this no-go theorem,
and the desirability of local contractibility for an efficient variational
ansatz, we provide guidelines for constructing networks consisting of multiple
types of tensors that can support power-law correlation. We also provide an
explicit construction of one such network, which approximates the holographic
HaPPY pentagon code in the limit where variational parameters are taken to be
small.
