  Holographic quantum error correcting codes (HQECC) have been proposed as toy
models for the AdS/CFT correspondence, and exhibit many of the features of the
duality. HQECC give a mapping of states and observables. However, they do not
map local bulk Hamiltonians to local Hamiltonians on the boundary. In this
work, we combine HQECC with Hamiltonian simulation theory to construct a
bulk-boundary mapping between local Hamiltonians, whilst retaining all the
features of the HQECC duality. This allows us to construct a duality between
models, encompassing the relationship between bulk and boundary energy scales
and time dynamics. It also allows us to construct a map in the reverse
direction: from local boundary Hamiltonians to the corresponding local
Hamiltonian in the bulk. Under this boundary-to-bulk mapping, the bulk geometry
emerges as an approximate, low-energy, effective theory living in the
code-space of an (approximate) HQECC on the boundary. At higher energy scales,
this emergent bulk geometry is modified in a way that matches the toy models of
black holes proposed previously for HQECC. Moreover, the duality on the level
of dynamics shows how these toy-model black holes can form dynamically.
