  One of the leading candidates for near-term quantum advantage is the class of
Variational Quantum Algorithms, but these algorithms suffer from classical
difficulty in optimizing the variational parameters as the number of parameters
increases. Therefore, it is important to understand the expressibility and
power of various ans\"atze to produce target states and distributions. To this
end, we apply notions of emulation to Variational Quantum Annealing and the
Quantum Approximate Optimization Algorithm (QAOA) to show that QAOA is
outperformed by variational annealing schedules with equivalent numbers of
parameters. Our Variational Quantum Annealing schedule is based on a novel
polynomial parameterization that can be optimized in a similar gradient-free
way as QAOA, using the same physical ingredients. In order to compare the
performance of ans\"atze types, we have developed statistical notions of
Monte-Carlo methods. Monte-Carlo methods are computer programs that generate
random variables that approximate a target number that is computationally hard
to calculate exactly. While the most well-known Monte-Carlo method is
Monte-Carlo integration (e.g. Diffusion Monte-Carlo or path-integral quantum
Monte-Carlo), QAOA is itself a Monte-Carlo method that finds good solutions to
NP-complete problems such as Max-cut. We apply these statistical Monte-Carlo
notions to further elucidate the theoretical framework around these quantum
algorithms.
