Complexity of Stoquastic Frustration-Free Hamiltonians
SIAM Journal on Computing
Quantum Information & Computation
The complexity of quantum spin systems on a two-dimensional square lattice
Quantum Information & Computation
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We study the computational strength of quantum particles (each of finite dimensionality) arranged on a line. First, we prove that it is possible to perform universal adiabatic quantum computation using a one-dimensional quantum system (with 9 states per particle).Building on the same construction, but with some additional technical effort and 12 states per particle, we show that the problem of approximating the ground state energy of a system composed of a line of quantum particles is QMA-complete; QMA is a quantum analogue of NP. This is in striking contrast to the analogous classical problem, one dimensional MAX-2-SAT with nearest neighbor constraints, which is in P. The proof of the QMA-completeness result requires an additional idea beyond the usual techniques in the area: Some illegal configurations cannot be ruled out by local checks, and are instead ruled out because they would, in the future, evolve into a state which can be seen locally to be illegal. Assuming BQP \ne QMA, our construction gives a one-dimensional system which takes an exponential time to relax to its ground state at any temperature. This makes it a candidate for a one-dimensional spin glass.