Quantum speedups for dynamic programming on n-dimensional lattice graphs

Motivated by the quantum speedup for dynamic programming on the Boolean hypercube by Ambainis et al. (2019), we investigate which graphs admit a similar quantum advantage. In this paper, we examine a generalization of the Boolean hypercube graph, the $n$-dimensional lattice graph $Q(D,n)$ with vertices in $\{0,1,\ldots,D\}^n$. We study the complexity of the following problem: given a subgraph $G$ of $Q(D,n)$ via query access to the edges, determine whether there is a path from $0^n$ to $D^n$. While the classical query complexity is $\widetilde{\Theta}((D+1)^n)$, we show a quantum algorithm with complexity $\widetilde O(T_D^n)$, where $T_D<D+1$. The first few values of $T_D$ are $T_1 \approx 1.817$, $T_2 \approx 2.660$, $T_3 \approx 3.529$, $T_4 \approx 4.421$, $T_5 \approx 5.332$. We also prove that $T_D \geq \frac{D+1}{\mathrm e}$, thus for general $D$, this algorithm does not provide, for example, a speedup, polynomial in the size of the lattice. While the presented quantum algorithm is a natural generalization of the known quantum algorithm for $D=1$ by Ambainis et al., the analysis of complexity is rather complicated. For the precise analysis, we use the saddle-point method, which is a common tool in analytic combinatorics, but has not been widely used in this field. We then show an implementation of this algorithm with time complexity $\text{poly}(n)^{\log n} T_D^n$, and apply it to the Set Multicover problem. In this problem, $m$ subsets of $[n]$ are given, and the task is to find the smallest number of these subsets that cover each element of $[n]$ at least $D$ times. While the time complexity of the best known classical algorithm is $O(m(D+1)^n)$, the time complexity of our quantum algorithm is $\text{poly}(m,n)^{\log n} T_D^n$.

由Ambainis等人在布尔超立方体上进行动态编程的量子加速动机。 （2019年），我们研究了哪些图形接受类似的量子优势。在本文中，我们研究了布尔式超立方体图的概括，即$ n $二维晶格图$ q（d，n）$，带有$ \ {0,1，\ ldots，d \}^n $的顶点。我们研究以下问题的复杂性：通过查询边缘的查询访问，给定$ q（d，n）$的子图$ g $，确定是否有$ 0^n $到$ d^n $的路径。虽然经典查询复杂性为$ \ widetilde {\ theta}（（（d+1）^n）$，但我们显示具有复杂性$ \ widetilde o（t_d^n）$的量子算法，其中$ t_d <d <d <d+1 $ 。 $ t_d $的前几个值是$ t_1 \大约1.817 $，$ t_2 \大约2.660 $，$ t_3 \大约3.529 $，$ t_4 \ of 4.421 $，$ t_5 \ of 5.332 $。我们还证明$ t_d \ geq \ frac {d+1} {\ mathrm e} $，因此对于一般$ d $，该算法不提供例如速度，以lattice的大小为多项式。尽管Ambainis等人对所提出的量子算法是对$ d = 1 $的已知量子算法的自然概括，但复杂性的分析相当复杂。对于精确的分析，我们使用鞍点方法，该方法是分析组合中的常见工具，但在该领域尚未广泛使用。然后，我们显示了具有时间复杂性$ \ text {poly}（n）^{\ log n} t_d^n $的实现，并将其应用于设置的多层问题。在此问题中，给出了$ [n] $的$ m $子集，任务是找到覆盖$ [n] $的每个元素的最小数量，至少$ d $ times。虽然最著名的古典算法的时间复杂性为$ O（m（d+1）^n）$，而我们的量子算法的时间复杂性为$ \ text {poly}（m，n）^{\ log n} t_d^n $。