  We introduce a new model of interacting spin 1/2. It describes interaction of
three nearest neighbors. The Hamiltonian can be expressed in terms of Fredkin
gates. The Fredkin gate (also known as the CSWAP gate) is a computational
circuit suitable for reversible computing. Our construction generalizes the
work of Ramis Movassagh and Peter Shor.
  Our model can be solved by means of Catalan combinatorics in the form of
random walks on the upper half of a square lattice [Dyck walks]. Each Dyck path
can be mapped to a wave function of the spins. The ground state is an equally
weighted superposition of Dyck walks [instead of Motzkin walks]. We can also
express it as a matrix product state. We further construct the model of
interacting spins 3/2 and greater half-integer spins. The models with higher
spins require coloring of Dyck walks. We construct SU(k) symmetric model [here
k is the number of colors]. The leading term of the entanglement entropy is
then proportional to the square root of the length of the lattice [like in
Shor-Movassagh model]. The gap closes as a high power of the length of the
lattice.
