Beating the random assignment on constraint satisfaction problems of bounded degree

We show that for any odd k and any instance of the Max-kXOR constraint satisfaction problem, there is an efficient algorithm that finds an assignment satisfying at least a frac12 + Omega(1/√D) fraction of constraints, where D is a bound on the number of constraints that each variable occurs in. This improves both qualitatively and quantitatively on the recent work of Farhi, Goldstone, and Gutmann (2014), which gave a quantum algorithm to find an assignment satisfying a frac12 + Omega(D-3/4) fraction of the equations. For arbitrary constraint satisfaction problems, we give a similar result for "triangle-free" instances; i.e., an efficient algorithm that finds an assignment satisfying at least a μ + Omega(1/√D) fraction of constraints, where μ is the fraction that would be satisfied by a uniformly random assignment.