一句话：提出ZX图加法的归纳定义，并据此给出ZX图的归纳微分，另提供基于变量隔离的非归纳微分及Ising哈密顿量图表示。
ZX演算是量子计算中一种有力的图形推理框架，能紧凑表示矩阵，但缺少对任意ZX图进行线性组合的形式化求和。本文借助受控图的构造，给出ZX图加法的归纳定义；基于该加法，进一步提出ZX图的归纳微分方法，可对角度描述中含变量的图按变量求导，并揭示微分与求和通过乘积规则紧密关联。文章还给出一种基于变量隔离的非归纳微分技术作为替代。最后，作者将结果应用于推导Ising哈密顿量对应的ZX图，说明这些操作可用于量子计算中的优化等问题。
The ZX-calculus is a powerful framework for reasoning in quantum computing. It provides in particular a compact representation of matrices of interests. A peculiar property of the ZX-calculus is the absence of a formal sum allowing the linear combinations of arbitrary ZX-diagrams. The universality of the formalism guarantees however that for any two ZX-diagrams, the sum of their interpretations can be represented by a ZX-diagram. We introduce a general, inductive definition of the addition of ZX-diagrams, relying on the construction of controlled diagrams. Based on this addition technique, we provide an inductive differentiation of ZX-diagrams. Indeed, given a ZX-diagram with variables in the description of its angles, one can differentiate the diagram according to one of these variables. Differentiation is ubiquitous in quantum mechanics and quantum computing (e.g. for solving optimization problems). Technically, differentiation of ZX-diagrams is strongly related to summation as witnessed by the product rules. We also introduce an alternative, non inductive, differentiation technique rather based on the isolation of the variables. Finally, we apply our results to deduce a diagram for an Ising Hamiltonian.
@article{paperbot2759,
title = {Addition and Differentiation of ZX-diagrams},
author = {Emmanuel Jeandel and Simon Perdrix and Margarita Veshchezerova},
journal = {Logical Methods in Computer Science},
volume = {Volume 20, Issue 2},
year = {2024},
doi = {10.46298/lmcs-20(2:10)2024}
}