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We study the computational strength of quantum particles (each of finitedimensionality) arranged on a line. First, we prove that it is possible toperform universal adiabatic quantum computation using a one-dimensional quantumsystem (with 9 states per particle). This might have practical implications forexperimentalists interested in constructing an adiabatic quantum computer.Building on the same construction, but with some additional technical effortand 12 states per particle, we show that the problem of approximating theground state energy of a system composed of a line of quantum particles isQMA-complete; QMA is a quantum analogue of NP. This is in striking contrast tothe fact that the analogous classical problem, namely, one-dimensionalMAX-2-SAT with nearest neighbor constraints, is in P. The proof of theQMA-completeness result requires an additional idea beyond the usual techniquesin the area: Not all illegal configurations can be ruled out by local checks,so instead we rule out such illegal configurations because they would, in thefuture, evolve into a state which can be seen locally to be illegal. Ourconstruction implies (assuming the quantum Church-Turing thesis and thatquantum computers cannot efficiently solve QMA-complete problems) that thereare one-dimensional systems which take an exponential time to relax to theirground states at any temperature, making them candidates for beingone-dimensional spin glasses.

The power of quantum systems on a line