Problem Solving using Quantum Computing and Quantum Clique Algorithm

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.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.

arXiv:1803.11356Physical Review AMar 29, 201817 citations

QUANTUM COMPUTING AND ALGORITHMS

Mr. K. Suresh Babu Dr. T. P. Anish Dr. Cornelius K

Mar 24, 20260 citations

Quantum computing and algorithms

Author 1 - Mr. K. Suresh Babu Author 2 : Dr. T. P. Anish Author 3 : Dr. Cornelius K

Dec 3, 20250 citations

A Comparison of Quantum Algorithms for the Maximum Clique Problem

Andrew Haverly

Two of the most promising computational models for quantum computing are the qubit-based model and the continuous variable model, which result in two different computational approaches, namely the qubit gate model and boson sampling. The qubit gate model is a universal form of quantum computation that relies heavily on the principles of superposition and entanglement to solve problems using qubits based on technologies ranging from magnetic fields created from superconducting materials to the spins of valence electrons in atoms. Boson sampling is a non-universal form of quantum computation that uses bosons as continuous-variable values for its computation. Both models show promising prospects for useful quantum advantages over classical computers, but these models are fundamentally different, not only on their technologies but on their applications. Each model excels in different sets of applications.\nA direct comparison for solving a problem using qubit gate models and boson sampling allows one to better understand not only the individual technologies, but how to decide which model is better suited to solving a given problem and how to start development on solving the given problem. This thesis uses the maximum clique problem to examine the application development process in the qubit gate model and boson sampling as well as a comparison of other known algorithms to the maximum clique problem. The maximum clique problem is an NP-Hard problem concerned with finding the largest fully-connected subgraph. The qubit gate model algorithm to the maximum clique problem is a novel algorithm.Two of the most promising computational models for quantum computing are the qubit-based model and the continuous variable model, which result in two different computational approaches, namely the qubit gate model and boson sampling. The qubit gate model is a universal form of quantum computation that relies heavily on the principles of superposition and entanglement to solve problems using qubits based on technologies ranging from magnetic fields created from superconducting materials to the spins of valence electrons in atoms. Boson sampling is a non-universal form of quantum computation that uses bosons as continuous-variable values for its computation. Both models show promising prospects for useful quantum advantages over classical computers, but these models are fundamentally different, not only on their technologies but on their applications. Each model excels in different sets of applications.\nA direct comparison for solving a problem using qubit gate models and boson sampling allows one to better understand not only the individual technologies, but how to decide which model is better suited to solving a given problem and how to start development on solving the given problem. This thesis uses the maximum clique problem to examine the application development process in the qubit gate model and boson sampling as well as a comparison of other known algorithms to the maximum clique problem. The maximum clique problem is an NP-Hard problem concerned with finding the largest fully-connected subgraph. The qubit gate model algorithm to the maximum clique problem is a novel algorithm.

Jan 1, 20213 citations

Quantum Algorithms

The MIT Press eBooksMar 19, 20190 citations

Circuit Synthesis of Maximum Clique Problem using Combinatorial Approach of Classical-Quantum Hybrid Model.

Arpita Sanyal, Amit Saha, Banani Saha +1 more

In the Maximum Clique Problem, the objective is to find a clique whose size is maximum among all the cliques of an arbitrary undirected and unweighted graph. Maximum Clique Problem resembles with minimum vertex cover problem, independent set problem. As it is an NP-hard problem, no polynomial time algorithms can be found. As these problems have several important practical applications such as information retrieval, community detection in network, spatial data mining etc, it is of great interest to try to synthesis the circuit of Maximum Clique Problem. In classical computing any brute-force solution to the Maximum Clique Problem requires an exponential increase of time with the size of the problem (i.e., with time complexity of $O(2^n)$ ). In this paper, Maximum Clique Problem has been solved using combinatorial approach of Grover's search algorithm. An algorithm has been proposed that auto-generates the circuit for any given undirected and unweighted graph which makes the approach generalized in nature for Maximum Clique Problem. Computational speed up is achieved by using our approach with the help of quantum mechanical effect that is quantum superposition. We have simulated the proposed circuit using IBM's QISkit platform and verified its correctness.In the Maximum Clique Problem, the objective is to find a clique whose size is maximum among all the cliques of an arbitrary undirected and unweighted graph. Maximum Clique Problem resembles with minimum vertex cover problem, independent set problem. As it is an NP-hard problem, no polynomial time algorithms can be found. As these problems have several important practical applications such as information retrieval, community detection in network, spatial data mining etc, it is of great interest to try to synthesis the circuit of Maximum Clique Problem. In classical computing any brute-force solution to the Maximum Clique Problem requires an exponential increase of time with the size of the problem (i.e., with time complexity of $O(2^n)$). In this paper, Maximum Clique Problem has been solved using combinatorial approach of Grover's search algorithm. An algorithm has been proposed that auto-generates the circuit for any given undirected and unweighted graph which makes the approach generalized in nature for Maximum Clique Problem. Computational speed up is achieved by using our approach with the help of quantum mechanical effect that is quantum superposition. We have simulated the proposed circuit using IBM's QISkit platform and verified its correctness.

Mar 10, 20200 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})

. 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.

O(2^n)

). In this paper, Maximum Clique Problem has been solved using combinatorial approach of Grover's search algorithm. An algorithm has been proposed that auto-generates the circuit for any given undirected and unweighted graph which makes the approach generalized in nature for Maximum Clique Problem. Computational speed up is achieved by using our approach with the help of quantum mechanical effect that is quantum superposition. We have simulated the proposed circuit using IBM's QISkit platform and verified its correctness.