We show that any two Hadamard graphs on the same number of vertices are quantum isomorphic. This follows from a more general recipe for showing quantum isomorphism of graphs arising from certain association schemes. The main result is built from three tools. A remarkable recent result [20] of Mančinska and Roberson shows that graphs G and H are quantum isomorphic if and only if, for any planar graph F, the number of graph homomorphisms from F to G is equal to the number of graph homomorphisms from F to H. A generalization of partition functions called “scaffolds” [23] affords some basic reduction rules such as series-parallel reduction and can be applied to counting homomorphisms. The final tool is the classical theorem of Epifanov showing that any plane graph can be reduced to a single vertex and no edges by extended series-parallel reductions and Delta-Wye transformations. This last sort of transformation is available to us in the case of exactly triply regular association schemes. The paper includes open problems and directions for future research.

我们证明，具有相同顶点数的任意两个哈达玛图都是量子同构的。这是从一个更通用的方法得出的，用于显示由某些关联方案产生的图的量子同构。主要结果是由三个工具构建的。Mančinska 和 Roberson最近的一项引人注目的结果[20]表明，图G和H是量子同构的，当且仅当对于任何平面图F ，从F到G的图同态数量等于从F的图同态数量到H . 称为“支架” [23]的配分函数的推广提供了一些基本的归约规则，例如串联并行归约，并且可以应用于计算同态。最后一个工具是 Epifanov 的经典定理，表明任何平面图都可以通过扩展串并联归约和 Delta-Wye 变换归约为单个顶点且没有边。在完全三重正则关联方案的情况下，我们可以使用最后一种转换。该论文包括未解决的问题和未来研究的方向。