The Complexity of Translationally-Invariant Spin Chains with Low Local Dimension

We prove that estimating the ground state energy of a translationallyinvariant, nearest-neighbour Hamiltonian on a 1D spin chain is QMAEXPcomplete, even for systems of low local dimension (≈ 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 very 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 significantly extend these techniques by allowing much 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. ∗Corresponding author: [email protected], +44 1223 760 237 †[email protected] ‡[email protected] ar X iv :1 60 5. 01 71 8v 2 [ qu an tph ] 1 9 Ja n 20 17