Quantum Annealing to Solve 3-Regular 3-XORSAT on a Lattice

Here we show how we can embed the 3-regular 3-XORSAT on a
square lattice made out of gates which couple the bits in a manner that
recreates the constraints. This system can be annealed to the solution
by tuning a transverse field to zero. We explore ways to avoid two
potential obstacles that limit how fast one can anneal this system: 1)
the nature of the phase transition as we tune the transverse field and
2) the avoided level crossings for small transverse field strength. We
discuss how the second pitfall can possibly be avoided in the lattice
embedding. We also present Quantum Monte Carlo results on the nature of
the phase transition for the embedding of XORSAT on the square lattice.
We compare the results to the embedding on a random regular graph, where
the phase transition is known to be first order.