一种新的受量子计算启发的Holant二分法
A new Holant dichotomy inspired by quantum computation
摘要：
Holant问题是分析图上计数复杂性问题的一个框架。这个框架是同时足够普遍，以涵盖许多其他计数问题的图和具体的足够允许二分法的结果的推导，划分所有的问题实例到那些可以解决的多项式时间和那些是#P-硬。Holant框架基于全息算法理论，该理论最初受到量子计算概念的启发，但这种联系似乎以前没有被探索过。在这里，我们采用量子信息理论来解释现有的结果在一个简洁的方式，并得出一个新的家庭的问题，我们称之为Holant $^+$的二分法。这个家族位于Holant $^*$的已知家族之间，对于Holant $^*$，已知完全二分法，而对于Holant $^c$，仅已知限制二分法。利用纠缠理论的知识-既有以前现有的工作和我们自己的新成果-我们证明了一个完整的二分法定理Holant $^+$，这是非常相似的限制Holant $^c$二分法。除了二分法#R $_3$ -CSP，我们是第一个Holant二分法，其中允许的功能是不受限制的，只有有限数量的功能被假定为是免费提供。
英文摘要：
Holant problems are a framework for the analysis of counting complexity problems on graphs. This framework is simultaneously general enough to encompass many other counting problems on graphs and specific enough to allow the derivation of dichotomy results, partitioning all problem instances into those which can be solved in polynomial time and those which are #P-hard. The Holant framework is based on the theory of holographic algorithms, which was originally inspired by concepts from quantum computation, but this connection appears not to have been explored before. Here, we employ quantum information theory to explain existing results in a concise way and to derive a dichotomy for a new family of problems, which we call Holant$^+$. This family sits in between the known families of Holant$^*$, for which a full dichotomy is known, and Holant$^c$, for which only a restricted dichotomy is known. Using knowledge from entanglement theory -- both previously existing work and new results of our own -- we prove a full dichotomy theorem for Holant$^+$, which is very similar to the restricted Holant$^c$ dichotomy. Other than the dichotomy for #R$_3$-CSP, ours is the first Holant dichotomy in which the allowed functions are not restricted and in which only a finite number of functions are assumed to be freely available.
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