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arXiv:2305.16644

Quantum Speedup for the Maximum Cut Problem

Weng-Long Chang, Renata Wong, Wen‐Yu Chung, Y. B. Chen, Ju-Chin Chen, Athanasios V. Vasilakos

Published May 26, 2023• 4 citations
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Abstract

Given an undirected, unweighted graph with $n$ vertices and $m$ edges, the maximum cut problem is to find a partition of the $n$ vertices into disjoint subsets $V_1$ and $V_2$ such that the number of edges between them is as large as possible. Classically, it is an NP-complete problem, which has potential applications ranging from circuit layout design, statistical physics, computer vision, machine learning and network science to clustering. In this paper, we propose a quantum algorithm to solve the maximum cut problem for any graph $G$ with a quadratic speedup over its classical counterparts, where the temporal and spatial complexities are reduced to, respectively, $O(\sqrt{2^n/r})$ and $O(m^2)$. With respect to oracle-related quantum algorithms for NP-complete problems, we identify our algorithm as optimal. Furthermore, to justify the feasibility of the proposed algorithm, we successfully solve a typical maximum cut problem for a graph with three vertices and two edges by carrying out experiments on IBM's quantum computer.

DOI
10.48550/arxiv.2305.16644

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Given an undirected, unweighted graph with nnn vertices and mmm edges, the maximum cut problem is to find a partition of the nnn vertices into disjoint subsets V1V_1V1​ and V2V_2V2​ such that the number of edges between them is as large as possible. Classically, it is an NP-complete problem, which has potential applications ranging from circuit layout design, statistical physics, computer vision, machine learning and network science to clustering. In this paper, we propose a quantum algorithm to solve the maximum cut problem for any graph GGG with a quadratic speedup over its classical counterparts, where the temporal and spatial complexities are reduced to, respectively, O(2n/r)O(\sqrt{2^n/r})O(2n/r​) and O(m2)O(m^2)O(m2). With respect to oracle-related quantum algorithms for NP-complete problems, we identify our algorithm as optimal. Furthermore, to justify the feasibility of the proposed algorithm, we successfully solve a typical maximum cut problem for a graph with three vertices and two edges by carrying out experiments on IBM's quantum computer.
Given an undirected, unweighted graph with n vertices and m edges, the maximum cut problem is to find a partition of the n vertices into disjoint subsetsV1{V}_{{1}}V1​andV2{V}_{{2}}V2​such that the number of edges between them is as large as possible. Classically, it is an NP-complete problem, which has potential applications ranging from circuit layout design, statistical physics, computer vision, machine learning and network science to clustering. In this paper, we propose a biomolecular and a quantum algorithm to solve the maximum cut problem for any graph G. The quantum algorithm is inspired by the biomolecular algorithm and has a quadratic speedup over its classical counterparts, where the temporal and spatial complexities are reduced to, respectively,O(2n/r){O}\text {(}\sqrt {{2}^{n}/{r}}\text {)}O(2n/r​)andO(m2){O}\text {(}{m}^{{2}}\text {)}O(m2). With respect to oracle-related quantum algorithms for NP-complete problems, we identify our algorithm as optimal. Furthermore, to justify the feasibility of the proposed algorithm, we successfully solve a typical maximum cut problem for a graph with three vertices and two edges by carrying out experiments on IBM’s quantum simulator.
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Graph partitioning is a fundamental problem in computer science with applications in diverse fields like network design, compiler optimization, and data analysis. Traditional algorithms for graph partitioning often suffer from computational complexity, particularly when dealing with large graphs. This paper explores the potential of utilizing quantum computing to accelerate graph partitioning. Specifically, we investigate the application of Grover's algorithm, a quantum search algorithm, to the problem. Our core claim is that by leveraging the superposition and interference capabilities inherent in quantum mechanics, a quantum solution can achieve a provable speedup compared to classical approaches. We present a theoretical framework outlining the algorithm's operation and derive the expected speedup. The key mathematical formulation involves representing the graph partitioning problem as an unstructured search problem amenable to Grover's algorithm. We define the problem as minimizing the cut size, represented by the objective function: *f(S) = Σi∈S Σj∈Sc d(i, j)* where *S* is a subset of vertices, *Sc* is the complement of *S* in the graph, and *d(i, j)* is the shortest path distance between vertices *i* and *j*. The algorithm then constructs a quantum oracle that maps the problem state to a superposition of all possible partitions. Grover's algorithm is then applied to amplify the amplitude of the solution state, resulting in a high probability of finding an optimal or near-optimal partition. The speedup is directly proportional to the square root of the number of iterations performed by Grover's algorithm, which is influenced by the problem size and the quality of the oracle. We analyze the complexity of the quantum algorithm and compare it to the complexity of the best-known classical algorithms, demonstrating a potential for significant speedup. Further research will focus on optimizing the oracle construction and exploring other quantum algorithms suitable for graph partitioning.
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The minimum cut problem in an undirected and weighted graph GGG is to find the minimum total weight of a set of edges whose removal disconnects GGG. We completely characterize the quantum query and time complexity of the minimum cut problem in the adjacency matrix model. If GGG has nnn vertices and edge weights at least 111 and at most τττ, we give a quantum algorithm to solve the minimum cut problem using O~(n3/2τ)\tilde O(n^{3/2}\sqrtτ)O~(n3/2τ​) queries and time. Moreover, for every integer 1≤τ≤n1 \le τ\le n1≤τ≤n we give an example of a graph GGG with edge weights 111 and τττ such that solving the minimum cut problem on GGG requires Ω(n3/2τ)Ω(n^{3/2}\sqrtτ)Ω(n3/2τ​) many queries to the adjacency matrix of GGG. These results contrast with the classical randomized case where Ω(n2)Ω(n^2)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 GGG has mmm edges the classical randomized complexity of the minimum cut problem is Θ~(m)\tilde Θ(m)Θ~(m). We show that the quantum query and time complexity are O~(mnτ)\tilde O(\sqrt{mnτ})O~(mnτ​) and O~(mnτ+n3/2)\tilde O(\sqrt{mnτ} + n^{3/2})O~(mnτ​+n3/2), respectively, where again the edge weights are between 111 and τττ. For dense graphs we give lower bounds on the quantum query complexity of Ω(n3/2)Ω(n^{3/2})Ω(n3/2) for τ>1τ> 1τ>1 and Ω(τn)Ω(τn)Ω(τn) for any 1≤τ≤n1 \leq τ\leq n1≤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).
An s-⁡ts{\operatorname{-}}ts-t minimum cut in a graph corresponds to a minimum weight subset of edges whose removal disconnects vertices sss and ttt. Finding such a cut is a classic problem that is dual to that of finding a maximum flow from sss to ttt. In this work we describe a quantum algorithm for the minimum s-⁡ts{\operatorname{-}}ts-t cut problem on undirected graphs. For an undirected graph with nnn vertices, mmm edges, and integral edge weights bounded by WWW, the algorithm computes with high probability the weight of a minimum s-⁡ts{\operatorname{-}}ts-t cut in time O~(mn5/6W1/3+n5/3W2/3)\widetilde O(\sqrt{m} n^{5/6} W^{1/3} + n^{5/3} W^{2/3})O(m​n5/6W1/3+n5/3W2/3), given adjacency list access to GGG. For simple graphs this bound is always O~(n11/6)\widetilde O(n^{11/6})O(n11/6), even in the dense case when m=Ω(n2)m = \Omega(n^2)m=Ω(n2). In contrast, a randomized algorithm must make Ω(m)\Omega(m)Ω(m) queries to the adjacency list of a simple graph GGG even to decide whether sss and ttt are connected.