  The Quantum Alternating Operator Ansatz (QAOA+) framework has recently gained
attention due to its ability to solve discrete optimization problems on noisy
intermediate-scale quantum (NISQ) devices in a manner that is amenable to
derivation of worst-case guarantees. We design a technique in this framework to
tackle a few problems over maximal matchings in graphs. Even though maximum
matching is polynomial-time solvable, most counting and sampling versions are
#P-hard.
  We design a few algorithms that generates superpositions over matchings
allowing us to sample from them. In particular, we get a superposition over all
possible matchings when given the empty state as input and a superposition over
all maximal matchings when given the W -states as input.
  Our main result is that the expected size of the matchings corresponding to
the output states of our QAOA+ algorithm when ran on a 2-regular graph is
greater than the expected matching size obtained from a uniform distribution
over all matchings. This algorithm uses a W -state as input and we prove that
this input state is better compared to using the empty matching as the input
state.
