The semidefinite programming (SDP) relaxation has proven to be extremely strong for many hard discrete optimization problems. This is in particular true for the quadratic assignment problem (QAP), arguably one of the hardest NP-hard discrete optimization problems. There are several difficulties that arise in efficiently solving the SDP relaxation, e.g.,~increased dimension; inefficiency of the current primal-dual interior point solvers in terms of both time and accuracy; and difficulty and high expense in adding cutting plane constraints. We propose using the alternating direction method of multipliers (ADMM) to solve the SDP relaxation. This first order approach allows for inexpensive iterations, a method of cheaply obtaining low rank solutions, as well a trivial way of adding cutting plane inequalities. When compared to current approaches and current best available bounds we obtain remarkable robustness, efficiency and improved bounds.