The Bayesian network structure learning (BNSL) problem asks for a directed acyclic graph that maximizes a given score function. For networks with $n$ nodes, the fastest known algorithms run in time $O(2^n n^2)$ in the worst case, with no improvement in the asymptotic bound for two decades. Inspired by recent advances in quantum computing, we ask whether BNSL admits a polynomial quantum speedup, that is, whether the problem can be solved by a quantum algorithm in time $O(c^n)$ for some constant $c$ less than $2$. We answer the question in the affirmative by giving two algorithms achieving $c \leq 1.817$ and $c \leq 1.982$ assuming the number of potential parent sets is, respectively, subexponential and $O(1.453^n)$. Both algorithms assume the availability of a quantum random access memory.

翻译：贝叶斯网络结构学习（BNSL）问题要求寻找一个最大化给定得分函数的有向无环图。对于包含$n$个节点的网络，当前已知最快的算法在最坏情况下的时间复杂度为$O(2^n n^2)$，且该渐进界在二十年内未有改进。受近期量子计算进展的启发，我们探究BNSL是否存在多项式程度的量子加速，即能否用量子算法在$O(c^n)$时间内解决该问题，其中常数$c$小于2。我们通过给出两种算法对此问题给出肯定回答：在潜在父节点集数量分别呈次指数增长和$O(1.453^n)$的条件下，这两种算法分别实现$c \leq 1.817$和$c \leq 1.982$。两种算法均假设存在量子随机存取存储器。