Variational quantum counting with the alternating operator ansatz

We introduce a variational algorithm based on the quantum alternating operator ansatz (QAOA) for the approximate solution of computationally hard counting problems. Our algorithm, dubbed VQCount, is based on the equivalence between random sampling and approximate counting and employs QAOA as a solution sampler. We first prove that VQCount improves upon previous work by reducing exponentially the number of samples needed to obtain an approximation within a multiplicative factor of the exact count. Using tensor network simulations, we then study the typical performance of VQCount with shallow circuits on synthetic instances of two #P-hard problems, positive #NAE3SAT and positive #1-in-3SAT. We employ the original quantum approximate optimization algorithm version of QAOA, as well as the Grover-mixer variant which guarantees a uniform solution probability distribution. We observe a tradeoff between QAOA success probability and sampling uniformity, which we exploit to achieve an exponential gain in efficiency over naive rejection sampling. Our results highlight the potential and limitations of hybrid variational algorithms for the approximate solution of counting problems.