2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/154980We prove that adiabatic computation is equivalent to standard quantum computation even when the adiabatic quantum system is restricted to be a set of particles on a one-dimensional chain. We give a construction that uses a 2-local Hamiltonian on nearest neighbors using particles that can have ten distinct states. This implies a construction of a one-dimensional chain of qubits in which the Hamiltonian is 6-local. We adapt this construction to show that the 2-local Hamiltonian for 13-state particles is QMA-complete which in turn implies that the 8-local Hamiltonian restricted to a one-dimensional chain of qubits is QMA-complete.This paper has been merged with arXiv:0705.4077 and is now co-authored with Dorit Aharonov, Daniel Gottesman, and Julia Kempe. Ther version posted here is the same as the original versionQuantum Physicstext