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Chinese Physics B

Quantum algorithm for minimum dominating set problem with circuit design

Haoying 皓颖 Zhang 张, Shaoxuan 绍轩 Wang 王, Xinjian 新建 Liu 刘, Yingtong 颖童 Shen 沈, Yukun 玉坤 Wang 王

Published Oct 13, 2023• 6 citations
Publisher

Abstract

Using quantum algorithms to solve various problems has attracted widespread attention with the development of quantum computing. Researchers are particularly interested in using the acceleration properties of quantum algorithms to solve NP-complete problems. This paper focuses on the well-known NP-complete problem of finding the minimum dominating set in undirected graphs. To expedite the search process, a quantum algorithm employing Grover’s search is proposed. However, a challenge arises from the unknown number of solutions for the minimum dominating set, rendering direct usage of original Grover’s search impossible. Thus, a swap test method is introduced to ascertain the number of iterations required. The oracle, diffusion operators, and swap test are designed with achievable quantum gates. The query complexity is O(1.414 n ) and the space complexity is O(n). To validate the proposed approach, qiskit software package is employed to simulate the quantum circuit, yielding the anticipated results.

DOI
10.1088/1674-1056/ad02e5

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Using quantum algorithms to solve various problems has attracted widespread attention with the development of quantum computing. Researchers are particularly interested in using the acceleration properties of quantum algorithms to solve NP-complete problems. This paper focuses on the well-known NP-complete problem of finding the minimum dominating set in undirected graphs. To expedite the search process, a quantum algorithm employing Grover’s search is proposed. However, a challenge arises from the unknown number of solutions for the minimum dominating set, rendering direct usage of original Grover’s search impossible. Thus, a swap test method is introduced to ascertain the number of iterations required. The oracle, diffusion operators, and swap test are designed with achievable quantum gates. The query complexity is O(1.414 n ) and the space complexity is O(n). To validate the proposed approach, qiskit software package is employed to simulate the quantum circuit, yielding the anticipated results.
Grover's algorithm is a well-known unstructured quantum search algorithm run on quantum computers. It constructs an oracle and calls the oracle O(N\sqrt NN​) times to locate specific data out of N unsorted data. This represents a quadratic speedup compared to the classical unstructured data sequential search algorithm, which requires to call the oracle O(N) times. We are currently in the noisy intermediate-scale quantum (NISQ) era in which quantum computers have a limited number of qubits, short decoherence time, and low gate fidelity. It is thus desirable to design quantum components with three good properties: (i) a reduced number of qubits, (ii) shorter quantum depth, and (iii) fewer gates. This paper utilizes novel quantum counters with the above-mentioned three good properties to construct the oracle of Grover's algorithm to efficiently solve the dominating set problem (DSP), as defined below. For a given graph G=(V, E), a dominating set (DS) D is a subset of the vertex set V, such that every vertex is in D or has an adjacent vertex in D. The DSP is to decide for a given graph G and an integer k whether there exists a DS with size k. Algorithms solving the DSP have many applications. For example, they can be applied to check whether k routers suffice to connect all computers in a computer network. The DSP is an NP-complete problem, indicating that no classical algorithm exists to solve the DSP with polynomial time complexity in the worst case. Therefore, using quantum algorithms, such as Grover's algorithm, to exploit the potent computational capabilities of quantum computers to solve the DSP is highly promising. We execute the whole quantum circuit of Grover's algorithm using novel quantum counters through the IBM Quantum Lab service to validate that the circuit can solve the DSP efficiently and correctly.
Abstract In recent years, quantum computing has gradually extended its influence beyond the realm of physics research into the fields of electrical engineering and computer science. Most researchers and programmers remain more familiar with traditional algorithmic techniques based on conventional computer architectures. To address this gap, this study proposes a quantum algorithmic circuit design framework with Grover’s search speedup technique and provides theoretical proofs for some design techniques, aiming to facilitate knowledge transfer and ease the learning curve for designers entering the field of quantum algorithm development. Since combinatorial optimization problems in graph theory serve as the foundation for many practical applications, this study adopts the well-known Connected Dominating Set (CDS) problem as a design example to illustrate the practical applicability of the proposed quantum algorithmic design framework and presents a quantum circuit as a potential solution for addressing realworld challenges in network optimization and related applications. In addition, the circuit proposed in this paper can serve as a quantum oracle to identify connected dominating sets (CDSs) of a graph. When the oracle is applied to a superposition of vertex subsets, Grover’s search algorithm achieves a quadratic speedup. We designed a method for adjusting the initial amplitudes so that the search can be biased toward smaller CDSs, which maximizes the probability of finding the minimum CDS.
Quantum computing offers an accelerated mechanism for solving combinatorial optimization problems that could be hardly solvable with classical tools, in this context, this paper highlights the utility of quantum computing in solving NP-complete problems efficiently and shows up the importance of employing quantum algorithms achieving parallel calculation in the purpose of reducing the algorithmic complexity and simplifying the development of optimization applications. Within this publication, an optimized version of quantum circuit conception for developing the oracle of the Grover's search algorithm is introduced for solving the vertex cover problem, and it was also tested using the Qiskit library and applied to a set of instances to evaluate its performance.
Nowadays in Quantum Computing, the implementation of quantum algorithm has created a stir since Noisy Intermediate-Scale Quantum (NISQ) devices are out in the market. Researchers are mostly interested in solving NP-complete problems with the help of quantum algorithms for its speedup. As per the work on computational complexity by Karp [1], if any of the NP-complete problem can be solved then any other NP-complete problem can be reduced to that problem in polynomial time. In this Paper, K-coloring problem (NP-complete problem) has been considered to solve using Grover's search. A comparator-based approach has been used to implement K-coloring problem which enables the reduction of the qubit cost compared to the state-of-the-art. An end-to-end automated framework has been proposed to implement the K-coloring problem for any unweighted and undirected graph on any available Noisy Intermediate-Scale Quantum (NISQ) devices, which helps in generalizing our approach.
Although NP-complete problems have solutions that can be verified in polynomial time, their exponential computational complexity presents significant challenges for classical computing methods. This study focuses on transferring various NP-complete problems into the quantum computing framework. The application of specific quantum computing algorithms is proposed for solving these problems. Within the scope of this study, a reusable quantum component with problem-specific constraints is designed, and a quantum circuit model is introduced to adapt the Quantum Feasibility Labeling algorithm for solving the Maximum Clique problem. Additionally, the Quick Quantum Search algorithm, which performs a search with a single oracle call, is integrated into the solution of the Maximum Independent Set problem. The proposed quantum circuits provide illustrative examples of how the components of an NP-complete problem can be represented in a quantum computing environment and how problem-specific constraints can be implemented within a quantum circuit. The utilization of modular circuit designs effectively demonstrates the applicability of larger and more complex problems in quantum environments. These approaches not only enable more efficient use of computational resources compared to traditional methods but also lead to significant reductions in solution times. This study provides a comprehensive framework that not only demonstrates the practicality of quantum circuits for small-scale NP-complete problems but also lays the groundwork for addressing larger and more complex challenges in future quantum computing applications.
Many applications such as Graph coloring, Triangle finding, Boolean satisfiability, Traveling salesman problem can be solved by Grover's search algorithm, which is one of the remarkable quantum computing algorithm. To design a quantum circuit for a given quantum algorithm, which involves Grover's search, we need to define an oracle circuit specific to the given algorithm and the diffusion operator for amplification of the desired quantum state. In this paper, we propose a quantum circuit implementation for the oracle of the vertex coloring problem based on Grover's algorithmic approach. To the best of our knowledge, this is a first of its kind approach in regards to the quantum circuit synthesis of the vertex coloring oracle in binary quantum domain. We have performed the synthesis of the proposed oracle circuit for the six commonly available Physical Machine Description (PMD) gate libraries using the Fault Tolerant Quantum Logic Synthesis (FTQLS) tool. The synthesis results have been presented to understand the cost estimation of the oracle circuit for the various PMDs in terms of number of quantum operations and number of cycles.