  Tensor networks implementing quantum error correcting codes have recently
been used to construct toy models of holographic duality explicitly realizing
some of the more puzzling features of the AdS/CFT correspondence. These models
reproduce the Ryu-Takayanagi entropy formula for boundary intervals, and allow
bulk operators to be mapped to the boundary in a redundant fashion. These
exactly solvable, explicit models have provided valuable insight but
nonetheless suffer from many deficiencies, some of which we attempt to address
in this article. We propose a new class of tensor network models that subsume
the earlier advances and, in addition, incorporate additional features of
holographic duality, including: (1) a holographic interpretation of all
boundary states, not just those in a "code" subspace, (2) a set of bulk states
playing the role of "classical geometries" which reproduce the Ryu-Takayanagi
formula for boundary intervals, (3) a bulk gauge symmetry analogous to
diffeomorphism invariance in gravitational theories, (4) emergent bulk locality
for sufficiently sparse excitations, and (5) the ability to describe geometry
at sub-AdS resolutions or even flat space.
