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Quantum algorithms for community detection and their empirical run-times

Cade, C.W. (Chris), Folkertsma, M.J. (Marten), Ido Niesen, Weggemans, J.R. (Jordi)

Published Apr 1, 2024• 2 citations
Publisher

Abstract

We apply recent work [9] on empirical estimates of quantum speedups to the practical task of community detection in complex networks. We design several quantum variants of a popular classical algorithm – the Louvain algorithm for community detection – and first study their complexities in the usual way, before analysing their complexities empirically across a variety of artificial and real inputs. We find that this analysis yields insights not available to us via the asymptotic analysis, further emphasising the utility in such an empirical approach. In particular, we observe that a complicated quantum algorithm with a large asymptotic speedup might not be the fastest algorithm in practice, and that a simple quantum algorithm with a modest speedup might in fact be the one that performs best. Moreover, we repeatedly find that overheads such as those arising from the need to amplify the success probabilities of quantum sub-routines such as Grover search can nullify any speedup that might have been suggested by a theoretical worst- or expected-case analysis.

DOI
10.26421/qic24.5-6-1

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We apply recent work [9] on empirical estimates of quantum speedups to the practical task of community detection in complex networks. We design several quantum variants of a popular classical algorithm – the Louvain algorithm for community detection – and first study their complexities in the usual way, before analysing their complexities empirically across a variety of artificial and real inputs. We find that this analysis yields insights not available to us via the asymptotic analysis, further emphasising the utility in such an empirical approach. In particular, we observe that a complicated quantum algorithm with a large asymptotic speedup might not be the fastest algorithm in practice, and that a simple quantum algorithm with a modest speedup might in fact be the one that performs best. Moreover, we repeatedly find that overheads such as those arising from the need to amplify the success probabilities of quantum sub-routines such as Grover search can nullify any speedup that might have been suggested by a theoretical worst- or expected-case analysis.
We apply our recent work on empirical estimates of quantum speedups to the practical task of community detection in complex networks. We design several quantum variants of a popular classical algorithm -- the Louvain algorithm for community detection -- and first study their complexities in the usual way, before analysing their complexities empirically across a variety of artificial and real inputs. We find that this analysis yields insights not available to us via the asymptotic analysis, further emphasising the utility in such an empirical approach. In particular, we observe that a complicated quantum algorithm with a large asymptotic speedup might not be the fastest algorithm in practice, and that a simple quantum algorithm with a modest speedup might in fact be the one that performs best. Moreover, we repeatedly find that overheads such as those arising from the need to amplify the success probabilities of quantum sub-routines such as Grover search can nullify any speedup that might have been suggested by a theoretical worst- or expected-case analysis.
We apply recent work~\cite{ourotherpaper} on empirical estimates of quantum speedups to the practical task of community detection in complex networks. We design several quantum variants of a popular classical algorithm -- the \textit{Louvain algorithm} for community detection -- and first study their complexities in the usual way, before analysing their complexities empirically across a variety of artificial and real inputs. We find that this analysis yields insights not available to us via the asymptotic analysis, further emphasising the utility in such an empirical approach. In particular, we observe that a complicated quantum algorithm with a large asymptotic speedup might not be the fastest algorithm in practice, and that a simple quantum algorithm with a modest speedup might in fact be the one that performs best. Moreover, we repeatedly find that overheads such as those arising from the need to amplify the success probabilities of quantum sub-routines such as Grover search can nullify any speedup that might have been suggested by a theoretical worst- or expected-case analysis.
Run-times of quantum algorithms are often studied via an asymptotic, worst-case analysis. Whilst useful, such a comparison can often fall short: it is not uncommon for algorithms with a large worst-case run-time to end up performing well on instances of practical interest. To remedy this it is necessary to resort to run-time analyses of a more empirical nature, which for sufficiently small input sizes can be performed on a quantum device or a simulation thereof. For larger input sizes, alternative approaches are required.In this paper we consider an approach that combines classical emulation with detailed complexity bounds that include all constants. We simulate quantum algorithms by running classical versions of the sub-routines, whilst simultaneously collecting information about what the run-time of the quantum routine would have been if it were run instead. To do this accurately and efficiently for very large input sizes, we describe an estimation procedure and prove that it obtains upper bounds on the true expected complexity of the quantum algorithms.We apply our method to some simple quantum speedups of classical heuristic algorithms for solving the well-studied MAX- k -SAT optimization problem. This requires rigorous bounds (including all constants) on the expected- and worst-case complexities of two important quantum sub-routines: Grover search with an unknown number of marked items, and quantum maximum-finding. These improve upon existing results and might be of broader interest.Amongst other results, we found that the classical heuristic algorithms we studied did not offer significant quantum speedups despite the existence of a theoretical per-step speedup. This suggests that an empirical analysis such as the one we implement in this paper already yields insights beyond those that can be seen by an asymptotic analysis alone.
The present study aims to examine the real-world implications of Grover's algorithm, specifically focusing on its quadratic speedup in contrast to classical linear search techniques employed for unsorted databases. The objective is to bridge the gap between theoretical potential and practical application in the field of quantum computing. This study conducts a comprehensive analysis of Grover's Algorithm and traditional linear search across databases of varying sizes, uniquely contributing by empirically validating Grover's Algorithm's superior performance and aligning it with theoretical expectations. Significantly, the study expands its analysis to encompass larger databases, further affirming the algorithm's utility in handling growing database sizes. This study significantly contributes to the progress of quantum computing through the practical implementation of theoretical principles as well as enhancing the scientific community's understanding of quantum search algorithms and their relative efficacy.
The dynamics and underlying structure of complex social networks (SNs) can be discovered using community detection. Considering how quantum computing might improve community detection techniques is becoming more and more popular in light of recent developments in computing technology. In this article, the rapidly developing topic of quantum-SN analysis for community discovery is thoroughly reviewed. Community detection is studied in the context of several quantum-inspired techniques, including quantum annealing and quantum-inspired optimization. To make use of the strengths of both conventional SN research methods and quantum computing techniques, hybrid quantum-classical approaches are being investigated. Case studies and applications that have made use of quantum-SN analysis methodologies are reviewed to highlight the practical consequences and potential advantages over conventional methods. The article also highlights the challenges and limitations of using quantum computing for SN analysis, including technical constraints and ethical issues. Finally, prospects and future research objectives in the area of quantum-SN analysis are highlighted. This covers possible developments in quantum algorithms for community detection, the incorporation of quantum computing with other SN research tasks, and the significance of multidisciplinary collaborations.
In a network, the vertices with similar characteristics construct communities. The vertices in a community are well-connected. Detecting the communities in a network is a challenging and important problem in the theory of complex networks. One approach to solving this problem uses the classical random walks on graphs. In quantum computing, quantum walks are the quantum mechanical counterparts of classical random walks. In this article, we employ a variant of Szegedy's quantum walk to develop a procedure for discovering the communities in networks. The limiting probability distribution of quantum walks assists us in determining the inclusion of a vertex in a community. We apply our community detection procedure to a variety of graphs and social networks, including the relaxed caveman graph, lll-partition graph, Karate club graph, and the dolphin's social network, among others.