  We formulate a conceptually new model in which quantum mechanics emerges from
classical mechanics. Given a local Hamiltonian $H$ acting on $n$ qubits, we
define a local classical model with an additional spatial dimension whose
boundary dynamics is approximately -- but to arbitrary precision -- described
by Schr\"{o}dinger's equation and $H$. The bulk consists of a lattice of
classical bits that propagate towards the boundary through a circuit of
stochastic matrices. The bits reaching the boundary are governed by a
probability distribution whose deviation from the uniform distribution can be
interpreted as the quantum-mechanical wavefunction. Bell nonlocality is
achieved because information can move through the bulk much faster than the
boundary speed of light. We analytically estimate how much the model deviates
from quantum mechanics, and we validate these estimates using computer
simulations.
