Tree Containment 问题的一种 QUBO 表述
A QUBO formulation for the Tree Containment problem
AI总结：
本文将量子计算与系统发育学结合，为一般情形下 NP 完全的 Tree Containment 问题提出 QUBO 表述，并证明其逻辑量子比特需求为 O(n_N n_T)。
摘要：
系统发育（进化）树和网络是带叶标签的图，被广泛用于表示物种、语言、癌细胞和病毒等实体之间的进化关系。为了重建和分析系统发育网络，判断一个给定的有根系统发育网络是否嵌入一个给定的有根系统发育树这一问题持续受到关注。该问题在形式上称为 Tree Containment，一般情况下是 NP 完全的，但对某些类别的系统发育网络可在多项式时间内求解。本文将量子计算与系统发育学的思想结合起来，针对一般情形下的 Tree Containment 提出一种高效的二次无约束二值优化（Quadratic Unconstrained Binary Optimization，QUBO）表述。对于 Tree Containment 的实例 (N,T)，其中 N 是具有 n_N 个顶点的系统发育网络，T 是具有 n_T 个顶点的系统发育树，我们的表述所需的逻辑量子比特数为 O(n_N n_T)。
英文摘要：
Phylogenetic (evolutionary) trees and networks are leaf-labeled graphs that are widely used to represent the evolutionary relationships between entities such as species, languages, cancer cells, and viruses. To reconstruct and analyze phylogenetic networks, the problem of deciding whether or not a given rooted phylogenetic network embeds a given rooted phylogenetic tree is of recurring interest. This problem, formally know as Tree Containment, is NP-complete in general and polynomial-time solvable for certain classes of phylogenetic networks. In this paper, we connect ideas from quantum computing and phylogenetics to present an efficient Quadratic Unconstrained Binary Optimization formulation for Tree Containment in the general setting. For an instance (N,T) of Tree Containment, where N is a phylogenetic network with n_N vertices and T is a phylogenetic tree with n_T vertices, the number of logical qubits that are required for our formulation is O(n_N n_T).
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