  Holographic quantum-error correcting codes are models of bulk/boundary
dualities such as the anti-de Sitter/conformal field theory (AdS/CFT)
correspondence, where a higher-dimensional bulk geometry is associated with the
code's logical degrees of freedom. Previous discrete holographic codes based on
tensor networks have reproduced the general code properties expected from
continuum AdS/CFT, such as complementary recovery. However, the boundary states
of such tensor networks typically do not exhibit the expected correlation
functions of CFT boundary states. In this work, we show that a new class of
exact holographic codes, extending the previously proposed hyperinvariant
tensor networks into quantum codes, produce the correct boundary correlation
functions. This approach yields a dictionary between logical states in the bulk
and the critical renormalization group flow of boundary states. Furthermore,
these codes exhibit a state-dependent breakdown of complementary recovery as
expected from AdS/CFT under small quantum gravity corrections.
