  We introduce a family of dual-unitary circuits in 1+1 dimensions which
constitute a discrete analog of conformal field theories. These circuits are
quantum cellular automata which are invariant under the joint action of Lorentz
and scale transformations. Dual unitaries are four-legged tensors which satisfy
the unitarity condition across the time as well as the space direction, a
property that makes the model mathematically tractable. Using dual unitaries
too, we construct tensor-network states for our 1+1 model, which are
interpreted as spatial slices of curved 2+1 discrete geometries, where the
metric distance is defined by the entanglement structure of the state,
following Ryu-Takayanagi's prescription. The dynamics of the circuit induces a
natural dynamics on these geometries, which we study for flat and anti-de
Sitter spaces, and in the presence or absence of matter. We observe that the
dynamics of spaces with two or more particles differs from that of zero or one,
suggesting the presence of black holes. But this contrasts with the fact that
the family of models appears to be non-chaotic. We observe that the dynamics of
a particle in such spaces strongly depends on the presence of other particles,
suggesting gravitational back-reaction.
