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We construct a family of time-independent nearest-neighbor Hamiltonianscoupling eight-state systems on a 1D ring that enables universal quantumcomputation. Hamiltonians in this family can achieve universality either bydriving a continuous-time quantum walk or by terminating an adiabaticalgorithm. In either case, the universality property can be understood asarising from an efficient simulation of a programmable quantum circuit. Usinggadget perturbation theory, one can demonstrate the same kind of universalityfor related Hamiltonian families acting on qubits in 2D. Our resultsdemonstrate that simulating 1D chains of spin-7/2 particles is BQP-hard, andindeed BQP-complete because the outputs of decision problems can be encoded inthe outputs of such simulations.

Universal quantum walks and adiabatic algorithms by 1D Hamiltonians