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.

中文翻译：

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将量子交替算符Ansatz应用于图匹配问题

量子交替算子Ansatz（QAOA +）框架由于能够解决最坏情况的保证而解决嘈杂的中级量子（NISQ）器件上的离散优化问题的能力，最近引起了人们的关注。我们在此框架中设计了一种技术来解决图形中最大匹配方面的一些问题。即使最大匹配是多项式时间可解的，但大多数计数和采样版本都是＃P-hard。我们设计了一些算法，这些算法在匹配上生成叠加，从而使我们能够从匹配中进行采样。特别地，当给定空状态作为输入时，我们得到所有可能匹配的叠加，当给定W状态作为输入时，我们得到所有最大匹配的叠加。我们的主要结果是，在2个正则图上运行时，与我们的QAOA +算法的输出状态相对应的匹配的预期大小大于从所有匹配上的均匀分布获得的预期匹配大小。该算法使用W状态作为输入，并且我们证明此输入状态比使用空匹配作为输入状态更好。