Hamiltonian quantum computer in one dimension

Quantum computation can be achieved by preparing an appropriate initial product state of qudits and then letting it evolve under a fixed Hamiltonian. The readout is made by measurement on individual qudits at some later time. This approach is called the Hamiltonian quantum computation and it includes, for example, the continuous-time quantum cellular automata and the universal quantum walk. We consider one spatial dimension and study the compromise between the locality $k$ and the local Hilbert space dimension $d$. For geometrically 2-local (i.e., $k=2$), it is known that $d=8$ is already sufficient for universal quantum computation but the Hamiltonian is not translationally invariant. As the locality $k$ increases, it is expected that the minimum required $d$ should decrease. We provide a construction of Hamiltonian quantum computer for $k=3$ with $d=5$. One implication is that simulating 1D chains of spin-2 particles is BQP-complete. Imposing translation invariance will increase the required $d$. For this we also construct another 3-local ($k=3$) Hamiltonian that is invariant under translation of a unit cell of two sites but that requires $d$ to be 8.