  We prove that estimating the ground state energy of a
translationally-invariant, nearest-neighbour Hamiltonian on a 1D spin chain is
QMAEXP-complete, even for systems of low local dimension (roughly 40). This is
an improvement over the best previously-known result by several orders of
magnitude, and it shows that spin-glass-like frustration can occur in
translationally-invariant quantum systems with a local dimension comparable to
the smallest-known non-translationally-invariant systems with similar
behaviour.
  While previous constructions of such systems rely on standard models of
quantum computation, we construct a new model that is particularly well-suited
for encoding quantum computation into the ground state of a
translationally-invariant system. This allows us to shift the proof burden from
optimizing the Hamiltonian encoding a standard computational model to proving
universality of a simple model.
  Previous techniques for encoding quantum computation into the ground state of
a local Hamiltonian allow only a linear sequence of gates, hence only a linear
(or nearly linear) path in the graph of all computational states. We extend
these techniques by allowing significantly more general paths, including
branching and cycles, thus enabling a highly efficient encoding of our
computational model. However, this requires more sophisticated techniques for
analysing the spectrum of the resulting Hamiltonian. To address this, we
introduce a framework of graphs with unitary edge labels. After relating our
Hamiltonian to the Laplacian of such a unitary labelled graph, we analyse its
spectrum by combining matrix analysis and spectral graph theory techniques.
