The ZX-calculus is a graphical language which allows for reasoning about suitably represented tensor networks - namely ZX-diagrams - in terms of rewrite rules. Here, we focus on problems which amount to exactly computing a scalar encoded as a closed tensor network. In general, such problems are #P-hard. However, there are families of such problems which are known to be in P when the dimension is below a certain value. By expressing problem instances from these families as ZX-diagrams, we see that the easy instances belong to the stabilizer fragment of the ZX-calculus. Building on previous work on efficient simplification of qubit stabilizer diagrams, we present simplifying rewrites for the case of qutrits, which are of independent interest in the field of quantum circuit optimisation. Finally, we look at the specific examples of evaluating the Jones polynomial and of counting graph-colourings. Our exposition further champions the ZX-calculus as a suitable and unifying language for studying the complexity of a broad range of classical and quantum problems.
翻译：ZX 计算器是一种图形语言,它允许以重写规则来推理有适当代表性的电压网络,即 ZX diagrams 。 在这里, 我们集中关注一些问题, 这些问题相当于精确地计算一个标标语编码成封闭的 Exronor 网络。 一般来说, 这些问题是 # P- hard 。 但是, 当维度低于一定值时, 这些问题在 P 中是已知的。 通过将这些家庭的问题实例表述为 ZX diagrams, 我们可以看到这些简单的例子属于ZX caluls 稳定器的碎片。 在以前关于高效率简化 qubit 稳定器图的工作的基础上, 我们为Qutrits 的情况简化了重写器, Qutrits 在量子电路优化领域是独立感兴趣的。 最后, 我们查看了评估琼斯 多边波和计图色的具体实例。 我们进一步推崇ZX caluus, 作为研究广泛古典和量问题的复杂性的合适和统一语言。
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