发表机构
Leiden Institute of Advanced Computer Science, Leiden University; University of Wisconsin-Madison（莱顿大学高级计算机科学研究所; 威斯康星大学麦迪逊分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究群模决策图，提出广义LIMDD框架，证明其在泡利群基础上可带来指数级简洁性改进，并给出唯一规范形式及多项式时间计算方法。
AI 中文摘要
约简有序决策图是固定长度词上函数的最小自动机：其节点为残差，约简过程即Myhill-Nerode商。我们研究当该商被一个群粗化时会发生什么。固定一个作用在残差上的群$G$，当两个节点的残差位于同一个$G$-轨道时合并它们，并在边上记录群元素。对于函数$\{0,1\}^n\to\mathbb{C}$且$G$为泡利群时，这即是局部可逆映射决策图（LIMDD）。我们取$G$为两参数族，该族由$C^{\leq k}R_q$（具有最多$k$个控制量子比特、阶为$q$的相位旋转）生成，并考虑含或不含比特翻转$X$两种情况。我们证明，与Pauli-LIMDD相比，这带来了指数级的简洁性改进，并完整确定了该族的简洁性顺序。分离对象是超图态，它们总能被该族的某个成员高效表示。我们解决了五个查询和八个变换的可处理性问题，这些问题在该族中保持不变，并显示出与Pauli-LIMDD相同的行为。我们给出了一个五规则约简系统，其规范形式对该族的每个成员都是唯一的，并证明该规范形式可在LIMDD大小的多项式时间内计算。我们进一步表明，当粗化超出（反）对角群时，计算最小规模的规范形式变为非局部问题，可能需要重建整个决策图。
英文摘要
A reduced ordered decision diagram is the minimal automaton of a function on words of fixed length: its nodes are the residuals, and reduction is the Myhill--Nerode quotient. We study what happens when that quotient is coarsened by a group. Fix a group $G$ acting on residuals, merge two nodes when their residuals lie in one $G$-orbit, and record the group element on the edge. For functions $\{0,1\}^n\to\mathbb{C}$ and $G$ the Pauli group this is the Local Invertible Map Decision Diagram. We take $G$ from the two-parameter family generated by $C^{\leq k}R_q$, the phase rotation of order $q$ with up to $k$ control qubits, with and without the bit flip $X$. We show that this gives exponential succinctness improvements compared to Pauli-LIMDD, and we determine the succinctness order of the family completely. The separating objects are hypergraph states, which can always be efficiently represented by some member of the family. We settle the tractability of five queries and eight transformations, which is invariant across the family, and shows the same behavior as Pauli-LIMDD. We give a five-rule reduction system whose normal forms are unique for every member of the family, and we show that this canonical form is computable in polynomial time in the size of the LIMDD. We show that, when coarsening beyond the (anti-)diagonal groups, the calculation of a minimal sized normal form turns out to be non-local and it might to rebuild the whole diagram.
