Problems based on the structure of graphs -- for example finding cliques, independent sets, or colourings -- are of fundamental importance in classical complexity. It is well motivated to consider similar problems about quantum graphs, which are an operator system generalisation of graphs. Defining well-formulated decision problems for quantum graphs faces several technical challenges, and consequently the connections between quantum graphs and complexity have been underexplored. In this work, we introduce and study the clique problem for quantum graphs. Our approach utilizes a well-known connection between quantum graphs and quantum channels. The inputs for our problems are presented as quantum channels induced by circuits, which implicitly determine a corresponding quantum graph. We also use this approach to reimagine the clique and independent set problems for classical graphs, by taking the inputs to be circuits of deterministic or noisy channels which implicitly determine confusability graphs. We show that, by varying the collection of channels in the language, these give rise to complete problems for the classes $\textsf{NP}$, $\textsf{MA}$, $\textsf{QMA}$, and $\textsf{QMA}(2)$. In this way, we exhibit a classical complexity problem whose natural quantisation is $\textsf{QMA}(2)$, rather than $\textsf{QMA}$, which is commonly assumed. To prove the results in the quantum case, we make use of methods inspired by self-testing. To illustrate the utility of our techniques, we include a new proof of the reduction of $\textsf{QMA}(k)$ to $\textsf{QMA}(2)$ via cliques for quantum graphs. We also study the complexity of a version of the independent set problem for quantum graphs, and provide preliminary evidence that it may be in general weaker in complexity, contrasting to the classical case where the clique and independent set problems are equivalent.

翻译：基于图结构的问题——例如寻找团、独立集或着色——在经典复杂性理论中具有基础重要性。考虑量子图（图的算子系统推广）的类似问题具有充分的动机。为量子图制定良定义的判定问题面临若干技术挑战，因此量子图与复杂性之间的联系尚未得到充分探索。在本工作中，我们引入并研究了量子图的团问题。我们的方法利用了量子图与量子信道之间的已知联系。问题输入由电路诱导的量子信道表示，这些信道隐式定义了相应的量子图。我们还采用此方法重新构想经典图的团与独立集问题，即输入为确定性或噪声信道（隐式定义混淆图）的电路。我们证明，通过改变语言中的信道集合，这些问题分别产生了类$\textsf{NP}$、$\textsf{MA}$、$\textsf{QMA}$和$\textsf{QMA}(2)$的完全问题。由此，我们展示了一个经典复杂性问题，其自然量子化为$\textsf{QMA}(2)$，而非通常假设的$\textsf{QMA}$。为证明量子情形中的结果，我们采用了受自测试启发的方法。为说明这些技术的实用性，我们给出了通过量子图的团将$\textsf{QMA}(k)$归约到$\textsf{QMA}(2)$的新证明。我们还研究了量子图独立集问题变体的复杂性，并提供了初步证据表明其复杂性可能普遍较弱，这与团问题和独立集问题等价的经典情形形成对比。