A Quantum Approximate Optimization Algorithm Applied to a Bounded   Occurrence Constraint Problem
  We apply our recent Quantum Approximate Optimization Algorithm to the combinatorial problem of bounded occurrence Max E3LIN2. The input is a set of linear equations each of which contains exactly three boolean variables and each equation says that the sum of the variables mod 2 is 0 or is 1. Every variable is in no more than D equations. A random string will satisfy 1/2 of the equations. We show that the level one QAOA will efficiently produce a string that satisfies $\left(\frac{1}{2} + \frac{1}{101 D^{1/2}\, l n\, D}\right)$ times the number of equations. A recent classical algorithm achieved $\left(\frac{1}{2} + \frac{constant}{D^{1/2}}\right)$. We also show that in the typical case the quantum computer will output a string that satisfies $\left(\frac{1}{2}+ \frac{1}{2\sqrt{3e}\, D^{1/2}}\right)$ times the number of equations. 
Edward Farhi, Jeffrey Goldstone, Sam Gutmann
quant-ph