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Fredkin Spin Chain

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.