Quantum algorithms for graph problems are considered, both in the adjacency matrix model and in an adjacency list-like array model. We give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, Strong Connectivity, Minimum Spanning Tree, and Single Source Shortest Paths. For example, we show that the query complexity of Minimum Spanning Tree is in $\Theta(n^{3/2})$ in the matrix model and in $\Theta(\sqrt{nm})$ in the array model, while the complexity of Connectivity is also in $\Theta(n^{3/2})$ in the matrix model but in $\Theta(n)$ in the array model. The upper bounds utilize search procedures for finding minima of functions under various conditions.
Quantum algorithms for graph problems are considered, both in the adjacency matrix model and in an adjacency list-like array model. We give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, Strong Connectivity, Minimum Spanning Tree, and Single Source Shortest Paths. For example, we show that the query complexity of Minimum Spanning Tree is in Θ(n3/2) in the matrix model and in Θ(nm) in the array model, while the complexity of Connectivity is also in Θ(n3/2) in the matrix model but in Θ(n) in the array model. The upper bounds utilize search procedures for finding minima of functions under various conditions.
The edge connectivity of a simple graph is the least number of edges whose removal disconnects the graph. We study the quantum query complexity of computing the edge connectivity of a simple graph with n vertices and m edges in both the adjacency matrix and adjacency array models. For classical randomized algorithms, Ω(n2) and Ω(m) lower bounds are known for these models, respectively, showing that the trivial algorithm is optimal in terms of query complexity for both cases. In contrast, we show upper bounds of O~(n3/2) and O~(mn) on the quantum query complexity of computing edge connectivity in the adjacency matrix and adjacency array models, respectively. Our upper bound is tight up to logarithmic factors for the adjacency matrix model, while for the adjacency array model the best quantum query lower bound we know is Ω(n). Our upper bounds follow by combining a lemma on the structure of near-minimum cuts due to Rubinstein, Schramm and Weinberg (ITCS 2018) with a quantum sparsification algorithm by Apers and de Wolf (FOCS 2020).
For a weighted graph, instead of edge connectivity the relevant notion is the weight of a minimum cut, i.e. the minimum total weight of a set of edges whose removal disconnects the graph. For weighted graphs we show that in the worst case Ω(n2) queries can be needed to compute the weight of a minimum cut by a quantum algorithm in both the adjacency matrix and adjacency array models, thus no quantum speedup is possible in this case.
The minimum cut problem in an undirected and weighted graph G is to find the minimum total weight of a set of edges whose removal disconnects G. We completely characterize the quantum query and time complexity of the minimum cut problem in the adjacency matrix model. If G has n vertices and edge weights at least 1 and at most τ, we give a quantum algorithm to solve the minimum cut problem using O~(n3/2τ) queries and time. Moreover, for every integer 1≤τ≤n we give an example of a graph G with edge weights 1 and τ such that solving the minimum cut problem on G requires Ω(n3/2τ) many queries to the adjacency matrix of G. These results contrast with the classical randomized case where Ω(n2) queries to the adjacency matrix are needed in the worst case even to decide if an unweighted graph is connected or not. In the adjacency array model, when G has m edges the classical randomized complexity of the minimum cut problem is Θ~(m). We show that the quantum query and time complexity are O~(mnτ) and O~(mnτ+n3/2), respectively, where again the edge weights are between 1 and τ. For dense graphs we give lower bounds on the quantum query complexity of Ω(n3/2) for τ>1 and Ω(τn) for any 1≤τ≤n. Our query algorithm uses a quantum algorithm for graph sparsification by Apers and de Wolf (FOCS 2020) and results on the structure of near-minimum cuts by Kawarabayashi and Thorup (STOC 2015) and Rubinstein, Schramm and Weinberg (ITCS 2018). Our time efficient implementation builds on Karger's tree packing technique (STOC 1996).
The quantum query complexity of subgraph-containment problems, which ask whether a given subgraph H is present in an input graph G, has been the subject of considerable study. However, even for relatively simple subgraphs, such as paths and cycles, a complete understanding of their query complexities remains elusive. In this work, we consider several variants of path- and cycle-containment problems in the adjacency matrix model, where we search for paths or cycles of constant length k. We compare the settings where the graphs are directed or undirected, where the goal is to detect or find the existence of a path/cycle, and where the path/cycle we are looking for has length exactly k, or at most k. We also consider several promise versions of these problems, where we suppose that the input graph has a certain structure. We characterize the relative difficulty of these variants of the path/cycle-containment problems, by relating them to one another using randomized reductions, and grouping them into equivalence classes.
When we restrict our attention to path-containment problems, we get a dichotomy result. Some of the path-containment problems can be solved using a linear number of queries, and all the others are equivalent to one another (and additionally to several cycle-containment problems) under randomized reductions, up to constant overhead. For the latter equivalence class, we prove a novel quantum-walk-based algorithm that achieves query complexity O(n3/2−αk), where αk∈Θ(c−k) and c=3+17/2≈1.33, beating the previous best upper bound O(n3/2) on its query complexity. We also provide a conditional lower bound based on the graph-collision problem, which implies that this equivalence class does not admit linear-query quantum algorithms unless graph collision admits an O(n) query algorithm.
Harry Buhrman et al gave an Omega(sqrt n) lower bound for monotone graph properties in the adjacency matrix query model. Their proof is based on the polynomial method. However for some properties stronger lower bounds exist. We give an Omega(n^{3/2}) bound for Graph Connectivity using Andris Ambainis' method, and an O(n^{3/2} log n) upper bound based on Grover's search algorithm. In addition we study the adjacency list query model, where we have almost matching lower and upper bounds for Strong Connectivity of directed graphs.
The minimum cut problem in an undirected and weighted graph G is to find the minimum total weight of a set of edges whose removal disconnects G. We completely characterize the quantum query and time complexity of the minimum cut problem in the adjacency matrix model. If G has n vertices and edge weights at least 1 and at most τ, we give a quantum algorithm to solve the minimum cut problem using Õ(n^{3/2}√{τ}) queries and time. Moreover, for every integer 1 ≤ τ ≤ n we give an example of a graph G with edge weights 1 and τ such that solving the minimum cut problem on G requires Ω(n^{3/2}√{τ}) queries to the adjacency matrix of G. These results contrast with the classical randomized case where Ω(n²) queries to the adjacency matrix are needed in the worst case even to decide if an unweighted graph is connected or not. In the adjacency array model, when G has m edges the classical randomized complexity of the minimum cut problem is ̃ Θ(m). We show that the quantum query and time complexity are Õ(√{mnτ}) and Õ(√{mnτ} + n^{3/2}), respectively, where again the edge weights are between 1 and τ. For dense graphs we give lower bounds on the quantum query complexity of Ω(n^{3/2}) for τ > 1 and Ω(τ n) for any 1 ≤ τ ≤ n. Our query algorithm uses a quantum algorithm for graph sparsification by Apers and de Wolf (FOCS 2020) and results on the structure of near-minimum cuts by Kawarabayashi and Thorup (STOC 2015) and Rubinstein, Schramm and Weinberg (ITCS 2018). Our time efficient implementation builds on Karger’s tree packing technique (STOC 1996).