arXiv:1803.11356 · Physical Review A

Quantum speedup in solving the maximal-clique problem

Weng-Long Chang, Qi Yu, Zhaokai Li, Jiahui Chen, Xinhua Peng, Mang Feng

Published Mar 29, 2018• 17 citations

Abstract

The maximal-clique problem, to find the maximally sized clique in a given graph, is classically an NP-complete computational problem, which has potential applications ranging from electrical engineering, computational chemistry, and bioinformatics to social networks. Here we develop a quantum algorithm to solve the maximal-clique problem for any graph $G$ with $n$ vertices with quadratic speedup over its classical counterparts, where the time and spatial complexities are reduced to, respectively, $O(\sqrt{{2}^{n}})$ and $O({n}^{2})$. With respect to oracle-related quantum algorithms for the NP-complete problems, we identify our algorithm as optimal. To justify the feasibility of the proposed quantum algorithm, we successfully solve a typical clique problem for a graph $G$ with two vertices and one edge by carrying out a nuclear magnetic resonance experiment involving four qubits.

DOI: 10.1103/physreva.97.032344

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arXiv:1803.11356 · Physical Review A

Quantum speedup in solving the maximal-clique problem

Weng-Long Chang, Qi Yu, Zhaokai Li, Jiahui Chen, Xinhua Peng, Mang Feng

Published Mar 29, 2018• 17 citations

arXiv HTML PDF

Abstract

G

with

n

vertices with quadratic speedup over its classical counterparts, where the time and spatial complexities are reduced to, respectively,

O(\sqrt{{2}^{n}})

and

O({n}^{2})

. With respect to oracle-related quantum algorithms for the NP-complete problems, we identify our algorithm as optimal. To justify the feasibility of the proposed quantum algorithm, we successfully solve a typical clique problem for a graph

G

with two vertices and one edge by carrying out a nuclear magnetic resonance experiment involving four qubits.

DOI: 10.1103/physreva.97.032344

Related papers

Quantum Speedup for the Maximum Cut Problem

Weng-Long Chang, Renata Wong, Wen‐Yu Chung +3 more

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.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.

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The maximum clique of an undirected graph is the largest subgraph in which an edge exists between every vertex. The maximum clique problem presents itself in various fields and finding a tractable algorithm to solve the problem is important. This paper introduces an oracle to solve the maximum clique problem using Grover’s quantum algorithm. This approach potentially solves the maximum clique problem in $O\left( {\left| V \right|\sqrt {{2^{\left| V \right|}}} } \right)$ time complexity rather than the current classical time complexity of O(2|V|) of the best known algorithm, where |V| is the number of vertices in the graph. The full circuit implementation is presented using the open-source Qiskit environment [1]. The paper also analyzes the growth of the hardware resources requirements with the number of vertices in the targeted graph.The maximum clique of an undirected graph is the largest subgraph in which an edge exists between every vertex. The maximum clique problem presents itself in various fields and finding a tractable algorithm to solve the problem is important. This paper introduces an oracle to solve the maximum clique problem using Grover’s quantum algorithm. This approach potentially solves the maximum clique problem in $O\left( {\left| V \right|\sqrt {{2^{\left| V \right|}}} } \right)$ time complexity rather than the current classical time complexity of O(2|V|) of the best known algorithm, where |V| is the number of vertices in the graph. The full circuit implementation is presented using the open-source Qiskit environment [1]. The paper also analyzes the growth of the hardware resources requirements with the number of vertices in the targeted graph.

IEEE Computer Society Annual Symposium on VLSIJul 1, 202116 citations

Quantum Circuit Design for Finding k-Cliques via Quantum Amplitude Amplification Strategies

Simone Perriello

The 𝑘-clique problem, which involves identifying complete subgraphs of size 𝑘 within a graph, is a fundamental challenge in combinatorial optimization with applications in network analysis, bioinformatics, and cryptography.As the number of nodes 𝑛 grows, classical algorithms become computationally intractable, motivating the exploration of quantum computing to address these limitations.This work introduces two quantum algorithms for solving the 𝑘-clique problem: one based on Quantum Amplitude Amplification (QAA) and another leveraging Quantum Walks (QW) over the Johnson graph 𝐽 (𝑛, 𝑘).By exploiting the regular structure of the Johnson graph, the QW approach achieves the same quadratic speedup of the QAA approach over classical algorithms, while remaining applicable to arbitrary undirected, unweighted graphs.We provide detailed quantum circuit designs for both approaches, analyzing their performance in terms of qubit count, gate count, and circuit depth under the NOT-CNOT-Toffoli + arbitrary rotations and Clifford + T gate sets.Compared to state-of-the-art quantum algorithms, our methods achieve significant improvements in scaling, particularly for dense graphs, with speedups ranging from 2 5 to 2 20 .The 𝑘-clique problem, which involves identifying complete subgraphs of size 𝑘 within a graph, is a fundamental challenge in combinatorial optimization with applications in network analysis, bioinformatics, and cryptography.As the number of nodes 𝑛 grows, classical algorithms become computationally intractable, motivating the exploration of quantum computing to address these limitations.This work introduces two quantum algorithms for solving the 𝑘-clique problem: one based on Quantum Amplitude Amplification (QAA) and another leveraging Quantum Walks (QW) over the Johnson graph 𝐽 (𝑛, 𝑘).By exploiting the regular structure of the Johnson graph, the QW approach achieves the same quadratic speedup of the QAA approach over classical algorithms, while remaining applicable to arbitrary undirected, unweighted graphs.We provide detailed quantum circuit designs for both approaches, analyzing their performance in terms of qubit count, gate count, and circuit depth under the NOT-CNOT-Toffoli + arbitrary rotations and Clifford + T gate sets.Compared to state-of-the-art quantum algorithms, our methods achieve significant improvements in scaling, particularly for dense graphs, with speedups ranging from 2 5 to 2 20 .

ACM International Conference on Computing Frontiers (CF)May 28, 20252 citations

G

with

n

vertices with quadratic speedup over its classical counterparts, where the time and spatial complexities are reduced to, respectively,

O(\sqrt{{2}^{n}})

and

O({n}^{2})

G

with two vertices and one edge by carrying out a nuclear magnetic resonance experiment involving four qubits.

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.

The k-plex model, which allows each vertex to miss connections with up to

k

neighbors, serves as a relaxation of the clique model. Its adaptability makes it more suitable for analyzing graphs from real-world applications, where noise and imperfect data are common and the stringent clique model is often impractical. The challenge of identifying maximum k-plex (MKP, an NP-hard problem) is gaining attention in fields such as social network analysis, community detection, terrorist network identification, and graph clustering. Recent research efforts have focused on optimizing the time complexity of MKP algorithms. The state-of-the-art has reduced the complexity from a trivial

O^{*}(2^{n})

O^{*}(c_{k}^{n})

, with

c_{k} > 1.94

for

k

> 3, where

n

denotes the number of vertices. In this paper, we demonstrate that MKP can be solved in

O^{*}(1.42^{n})

and propose the first two quantum algorithms, qTKP and qMKP, to achieve this complexity. qTKP employs quantum search integrated with graph encoding, degree count, degree comparison, and size determination to find a k-plex of a given size; qMKP uses a binary search to progressively identify the maximum solution. To validate the practical performance and effectiveness of our algorithms, proof-of-principle experiments were conducted using the latest IBM quantum simulator currently available. This work holds potential to be applied to a wide range of clique relaxations, e.g., n-clan and n-club.

\boldsymbol{n}

-fold improvement over state-of-the-art Grover-based methods, which require

O(n\sqrt{2^n})

iterations and

O(n)

measurements for

n

-vertex graphs. We validate algorithmic correctness through simulations on IBM's Qiskit platform and benchmark qubit/gate efficiency against existing Grover-based MCP solvers.

O\left( {\left| V \right|\sqrt {{2^{\left| V \right|}}} } \right)

time complexity rather than the current classical time complexity of O(2|V|) of the best known algorithm, where |V| is the number of vertices in the graph. The full circuit implementation is presented using the open-source Qiskit environment [1]. The paper also analyzes the growth of the hardware resources requirements with the number of vertices in the targeted graph.