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arXiv:2004.10596 · IET Quantum Communication

Circuit design for clique problem and its implementation on quantum computer

Arpita Sanyal, Amit Saha, Banani Saha, Amlan Chakrabarti

Published Dec 9, 2021• 8 citations
arXivHTMLPDF

Abstract

Abstract Finding cliques in a graph has a wide range of applications due to its pattern matching ability. The k ‐clique problem, a subset of the clique problem, determines whether or not an arbitrary network has a clique of size k . Modern‐day applications include a variation of the k ‐clique problem that lists all cliques of size k . However, the quantum implementation of such a variation of the k ‐clique problem has not been addressed yet. In this work, apart from the theoretical solution of such a k ‐clique problem, practical quantum‐gate‐based implementation has been addressed using Grover's algorithm. In a classical‐quantum hybrid architecture, this approach is extended to build the circuit for the maximum clique problem. Our technique is generalised since the program automatically builds the circuit for any given undirected and unweighted graph and any chosen k . For a small k with regard to a big graph, the proposed solution to addressing the k ‐clique issue has shown a reduction in qubit cost and circuit depth when compared to the state‐of‐the‐art approach. A framework is also presented for mapping the automated generated circuit for clique problems to quantum devices. Using IBM's Qiskit, an analysis of the experimental results is demonstrated.

DOI
10.1049/qtc2.12029

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Abstract Finding cliques in a graph has a wide range of applications due to its pattern matching ability. The k ‐clique problem, a subset of the clique problem, determines whether or not an arbitrary network has a clique of size k . Modern‐day applications include a variation of the k ‐clique problem that lists all cliques of size k . However, the quantum implementation of such a variation of the k ‐clique problem has not been addressed yet. In this work, apart from the theoretical solution of such a k ‐clique problem, practical quantum‐gate‐based implementation has been addressed using Grover's algorithm. In a classical‐quantum hybrid architecture, this approach is extended to build the circuit for the maximum clique problem. Our technique is generalised since the program automatically builds the circuit for any given undirected and unweighted graph and any chosen k . For a small k with regard to a big graph, the proposed solution to addressing the k ‐clique issue has shown a reduction in qubit cost and circuit depth when compared to the state‐of‐the‐art approach. A framework is also presented for mapping the automated generated circuit for clique problems to quantum devices. Using IBM's Qiskit, an analysis of the experimental results is demonstrated.
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(2n)O(2^n)O(2n)). 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.
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 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(∣V∣2∣V∣)O\left( {\left| V \right|\sqrt {{2^{\left| V \right|}}} } \right)O(∣V∣2∣V∣​) 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.
Abstract The maximum clique problem (MCP) is to find the largest complete subgraph in an undirected graph, that is, the subgraph in which there are edges between every two different vertices. It is an NP-Hard problem with wide applications, including bioinformatics, social networks, data mining, and other fields. This paper proposes an enhanced quantum solution method that introduces a pre-detection phase based on a hybrid heuristic algorithm and a vertex pruning strategy, encoding the prior constraints on clique size into quantum registers as global information. It encodes the prior constraints on clique size into quantum registers as global information. In contrast, state-of-the-art Grover-based methods for n-vertex graphs require O ( n 2 n ) iterations and O ( n ) measurements. By leveraging a global information encoding mechanism, the proposed method compresses the number of iterations to O ( 2 n ) and only needs a constant number of measurements to identify all maximum cliques. We validate algorithmic correctness through simulations on IBM’s Qiskit platform and benchmark qubit/gate efficiency against existing Grover-based MCP solvers.
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.
Determining the cardinality of the maximum clique on a simple graph is a NP-hard problem with several applications of interest. The best know classical algorithm for this problem has 𝒪(3𝑛/3) time complexity. Classical computing uses phenomena from classical physics to model, store and process information. Consequently, the entire model of classical computing is bounded superiorly by the limits of classical physics. As an alternative, Quantum Computing has been developed to take advantage of quantum phenomena and provide the constructions of circuits capable of performing certain computations more efficiently. In this paper, two quantum solutions to the maximum clique problem will be proposed and analyzed, based on the Grover Quantum Search algorithms and the Quantum Minimal Search Algorithm, comparing their respective time complexities with the best known classical solution.