  We present a general procedure for constructing tensor networks that
accurately reproduce holographic states in conformal field theories (CFTs).
Given a state in a large-$N$ CFT with a static, semiclassical gravitational
dual, we build a tensor network by an iterative series of approximations that
eliminate redundant degrees of freedom and minimize the bond dimensions of the
resulting network. We argue that the bond dimensions of the tensor network will
match the areas of the corresponding bulk surfaces. For 'tree' tensor networks
(i.e., those that are constructed by discretizing spacetime with
non-intersecting Ryu-Takayanagi surfaces), our arguments can be made rigorous
using a version of one-shot entanglement distillation in the CFT. Using the
known quantum error correcting properties of AdS/CFT, we show that bulk legs
can be added to the tensor networks to create holographic quantum error
correcting codes. These codes behave similarly to previous holographic tensor
network toy models, but describe actual bulk excitations in continuum AdS/CFT.
  By assuming some natural generalizations of the 'holographic entanglement of
purification' conjecture, we are able to construct tensor networks for more
general bulk discretizations, leading to finer-grained networks that partition
the information content of a Ryu-Takayanagi surface into tensor-factorized
subregions. While the granularity of such a tensor network must be set larger
than the string/Planck scales, we expect that it can be chosen to lie well
below the AdS scale. However, we also prove a no-go theorem which shows that
the bulk-to-boundary maps cannot all be isometries in a tensor network with
intersecting Ryu-Takayanagi surfaces.
