  The quantum approximate optimization algorithm (QAOA) is a promising quantum
algorithm that can be used to approximately solve combinatorial optimization
problems. The usual QAOA ansatz consists of an alternating application of the
cost and mixer Hamiltonians. In this work, we study how using Hamiltonians
other than the usual cost Hamiltonian, dubbed custom phase operators, can
affect the performance of QAOA. We derive an expected value formula for QAOA
with custom phase operators at p = 1 and show numerically that some of these
custom phase operators can achieve higher approximation ratio than the original
algorithm implementation. Out of all the graphs tested, 0.036\% of the random
custom phase operators, 75.9\% of the subgraph custom phase operators, 95.1\%
of the triangle-removed custom phase operators, and 93.9\% of the maximal
degree edge-removed custom phase operators have a higher approximation ratio
than the original QAOA implementation. This finding opens up the question of
whether better phase operators can be designed to further improve the
performance of QAOA.
