ZX-演算：Clifford+T量子力学的分圆补与不完全性
ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T quantum mechanics
摘要：
ZX演算是量子力学和量子信息处理的一种强大的图形语言。语言的完整性--即导出任何真实方程的能力--是一个至关重要的问题。在寻求完整的ZX演算时，最近已经证明了平行性对于量子图推理是必要的（MFCS 2016）。粗略地说，平行性包括合并两个子图时，他们被参数化的对跖角。我们介绍了一个广义的cyclo-tomic cycloentarity-它包括在合并n个子图一次，当n个角度划分成相等的部分圆。我们证明了当n是奇素数时，分圆连续性是不能导出的，从而为图解量子推理提供了一个可数的新公理族，我们展示了另一个不能从ZX演算的现有规则导出的新的简单公理，特别是暗示了Clifford+T量子力学语言的不完备性.我们最终得到了一个新的公理化的扩展ZX-演算，包括一个公理模式的分圆的concurrentarity。
英文摘要：
The ZX-Calculus is a powerful graphical language for quantum mechanics and quantum information processing. The completeness of the language -- i.e. the ability to derive any true equation -- is a crucial question. In the quest of a complete ZX-calculus, supplementarity has been recently proved to be necessary for quantum diagram reasoning (MFCS 2016). Roughly speaking, supplementarity consists in merging two subdiagrams when they are parameterized by antipodal angles. We introduce a generalised supplementarity -- called cyclotomic supplementarity -- which consists in merging n subdiagrams at once, when the n angles divide the circle into equal parts. We show that when n is an odd prime number, the cyclotomic supplementarity cannot be derived, leading to a countable family of new axioms for diagrammatic quantum reasoning.We exhibit another new simple axiom that cannot be derived from the existing rules of the ZX-Calculus, implying in particular the incompleteness of the language for the so-called Clifford+T quantum mechanics. We end up with a new axiomatisation of an extended ZX-Calculus, including an axiom schema for the cyclotomic supplementarity.
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