Quantum Algorithms for Matching Problems

Kamil Ravilevich Khadiev, Danil Serov

Toru Fujimura

Margaret Gao, Rachel Huang, Arul Rhik Mazumder +1 more

In this paper, we study the string matching problem. We design a quantum string-matching algorithm for noisy intermediate-scale quantum (NISQ) computers, given the current leading quantum processing units (QPUs) having no more than a few hundred qubits [16]. We also compare the performance of classic algorithms and quantum algorithms under various combinations. Our study provides a comprehensive and quantitative guide for users to choose appropriate classic or quantum algorithms for their string matching problems.

Andris Ambainis, Robert Špalek

Jinyuan Dong, yao zhao, Zhiqiang Li +3 more

Abstract The three-dimensional matching problem is a classic NP-complete problem in combinatorial optimization, with wide applications in resource allocation, task scheduling, and many other practical scenarios. Traditional algorithms for solving the three-dimensional matching problem face the challenge of high time complexity, making it difficult to solve large-scale problems within a reasonable time frame. To improve the solution efficiency, a quantum circuit solution based on the Quantum Approximate Optimization Algorithm (QAOA) is proposed. By constructing a mathematical model of the three-dimensional matching problem, the corresponding Hamiltonian expression is derived. A QAOA-based quantum circuit is designed and the parameters of the parameterized quantum gates are optimized using the classical optimization algorithm COBYLA. Simulation experiments are performed using IBM's quantum development framework Qiskit. The experimental results show that the solution to the three-dimensional matching problem can be obtained with 84% probability in polynomial time, verifying the feasibility and effectiveness of solving the three-dimensional matching problem based on the Quantum Approximate Optimization Algorithm.

Tatsuya Terao, Ryuhei Mori

In this paper, we consider the parameterized quantum query complexity for graph problems. We design parameterized quantum query algorithms for k-vertex cover and k-matching problems, and present lower bounds on the parameterized quantum query complexity. Then, we show that our quantum query algorithms are optimal up to a constant factor when the parameters are small. Our main results are as follows. Parameterized quantum query complexity of vertex cover. In the k-vertex cover problem, we are given an undirected graph G with n vertices and an integer k, and the objective is to determine whether G has a vertex cover of size at most k. We show that the quantum query complexity of the k-vertex cover problem is O(√kn + k^{3/2}√n) in the adjacency matrix model. For the design of the quantum query algorithm, we use the method of kernelization, a well-known tool for the design of parameterized classical algorithms, combined with Grover’s search. Parameterized quantum query complexity of matching. In the k-matching problem, we are given an undirected graph G with n vertices and an integer k, and the objective is to determine whether G has a matching of size at least k. We show that the quantum query complexity of the k-matching problem is O(√kn + k²) in the adjacency matrix model. We obtain this upper bound by using Grover’s search carefully and analyzing the number of Grover’s searches by making use of potential functions. We also show that the quantum query complexity of the maximum matching problem is O(√pn + p²) where p is the size of the maximum matching. For small p, it improves known bounds Õ(n^{3/2}) for bipartite graphs [Blikstad-v.d.Brand-Efron-Mukhopadhyay-Nanongkai, FOCS 2022] and O(n^{7/4}) for general graphs [Kimmel-Witter, WADS 2021]. Lower bounds on parameterized quantum query complexity. We also present lower bounds on the quantum query complexities of the k-vertex cover and k-matching problems. The lower bounds prove the optimality of the above parameterized quantum query algorithms up to a constant factor when k is small. Indeed, the quantum query complexities of the k-vertex cover and k-matching problems are both Θ(√k n) when k = O(√n) and k = O(n^{2/3}), respectively.