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Advances in Quantum Computing and Applications
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Advances in Quantum Computing and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E1: Mathematics and Computer Science".

Deadline for manuscript submissions: 31 July 2025 | Viewed by 23937

Special Issue Editors

Prof. Dr. Frank Phillipson

E-Mail Website
Guest Editor
1. Netherlands Organisation for Applied Scientific Research (TNO), P.O. Box 96800 The Hague, The Netherlands
2. Computational Operations Research - Quantum Enhanced Decision Intelligence, Maastricht University, P.O. Box 616 Maastricht, The Netherlands
Interests: optimization; operations research; machine learning; quantum computing applications
Dr. Sebastian Feld

E-Mail Website
Guest Editor
Quantum & Computer Engineering Department, Delft University of Technology, 2628 XE Delft, The Netherlands
Interests: applied machine learning; heuristic optimization; quantum computer science; spatial intelligence; time series analysis; IT security
Dr. Matthias Möller

E-Mail Website
Guest Editor
Department of Applied Mathematics, Faculty of Electrical Engineering, Mathematics, and Computer Science, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands
Interests: classical and quantum algorithms for fluid dynamic applications; high-performance computing; quantum-accelerated numerical linear algebra; numerical methods for simulation and optimization
Special Issues, Collections and Topics in MDPI journals
Ward van der Schoot

E-Mail Website
Guest Editor Assistant
Netherlands Organisation for Applied Scientific Research (TNO), P.O. Box 96800 The Hague, The Netherlands
Interests: quantum algorithms and complexity; quantum machine learning; applications of gate-based quantum computing; applications of quantum annealing; benchmarking quantum devices
Niels Neumann

E-Mail Website
Guest Editor Assistant
1. Netherlands Organisation for Applied Scientific Research (TNO), P.O. Box 96800 The Hague, The Netherlands
2. Research center for Quantum software & technology (QuSoft), 1098 XG Amsterdam, The Netherlands
Interests: quantum algorithms and complexity; shallow (constant depth) circuit NISQ algorithms; quantum machine learning (QML); distributed quantum computing; secure cloud-based quantum computing

Special Issue Information

Dear Colleagues,

With the field of quantum computing maturing and the increasing discovery of application areas, it is expected that quantum computers will solve specific problems much faster than current and future generations of classical computers. The development of quantum computing hardware has seen significant progress in recent years from gate-based to adiabatic (quantum annealing) and photonic computational models. The first results confirm the feasibility of applying such hardware to real-world situations such as traffic flow optimization, drug discovery, portfolio optimization, encryption, and machine learning. It is important for the research community to highlight current progress in this field and encourage appropriate steps to be taken in the industry, and this Special Issue provides the platform to do so.

Prof. Dr. Frank Phillipson
Dr. Sebastian Feld
Dr. Matthias Möller
Guest Editors

Ward van der Schoot 
Niels Neumann
Guest Editor Assistants

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • advances in variational algorithms
  • applications of gate-based quantum computing
  • applications of photonic quantum computing
  • applications of quantum annealing
  • distributed quantum computing
  • error-correction algorithms
  • hybrid quantum computing
  • impact of quantum computing on business and society
  • implementation of quantum computing
  • new nisq algorithms and applications
  • performance evaluation of quantum algorithms
  • quantum algorithm structures and patterns
  • quantum algorithms and complexity
  • quantum hardware advances
  • Quantum machine learning (QML)
  • secure cloud-based quantum computing
  • shallow (constant-depth) algorithms
  • simulation of quantum processes

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Published Papers (10 papers)

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22 pages, 1962 KiB  
Article
Quantum-Fuzzy Expert Timeframe Predictor for Post-TAVR Monitoring
by Lilia Tightiz and Joon Yoo
Mathematics 2024, 12(17), 2625; https://doi.org/10.3390/math12172625 - 24 Aug 2024
Cited by 1 | Viewed by 848
Abstract
This paper presents a novel approach to predicting specific monitoring timeframes for Permanent Pacemaker Implantation (PPMI) requirements following a Transcatheter Aortic Valve Replacement (TAVR). The method combines Quantum Ant Colony Optimization (QACO) with the Adaptive Neuro-Fuzzy Inference System (ANFIS) and incorporates expert knowledge. [...] Read more.
This paper presents a novel approach to predicting specific monitoring timeframes for Permanent Pacemaker Implantation (PPMI) requirements following a Transcatheter Aortic Valve Replacement (TAVR). The method combines Quantum Ant Colony Optimization (QACO) with the Adaptive Neuro-Fuzzy Inference System (ANFIS) and incorporates expert knowledge. Although this forecast is more precise, it requires a larger number of predictors to achieve this level of accuracy. Our model deploys expert-derived insights to guarantee the clinical relevance and interpretability of the predicted outcomes. Additionally, we employ quantum computing techniques to address this complex and high-dimensional problem. Through extensive assessments, we show that our quantum-enhanced model outperforms traditional methods with notable improvement in evaluation metrics, such as accuracy, precision, recall, and F1 score. Furthermore, with the integration of eXplainable AI (XAI) methods, our solution enhances the transparency and reliability of medical predictive models, hence promoting improved clinical practice decision-making. The findings highlight how quantum computing has the potential to completely transform predictive analytics in the medical field, especially when it comes to improving patient care after TAVR. Full article
(This article belongs to the Special Issue Advances in Quantum Computing and Applications)
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Figure 1

Figure 1
<p>An overview of the system.</p> Full article ">Figure 2
<p>Flowchart of the ANFIS parameter optimization process via quantum-enhanced ACO for TAVR outcome prediction.</p> Full article ">Figure 3
<p>Graph overview of customized ANFIS-ACO network for enhanced PPMI prediction post-TAVR.</p> Full article ">Figure 4
<p>Benchmark model performance AUC evaluation comparison and convergence performance. (<b>a</b>) ROC curve and ROC-AUC of benchmark ML algorithms on test dataset. (<b>b</b>) Benchmark algorithms training convergence based on RMSE in each iteration.</p> Full article ">Figure 5
<p>Run time and SD of benchmark algorithms with different number of features.</p> Full article ">Figure 6
<p>Membership degree for baseline RBBB and age in the trained ANFIS Model. (<b>a</b>) Membership degree for baseline RBBB. (<b>b</b>) Membership degree for age.</p> Full article ">Figure 7
<p>Visualization of important features in decision-making with the proposed model through XAI techniques. (<b>a</b>) Impact of features on the proposed model output prediction based on SHAP values. (<b>b</b>) Impact of features on model output prediction for an individual sample based on LIME values.</p> Full article ">
17 pages, 7681 KiB  
Article
A Modified Depolarization Approach for Efficient Quantum Machine Learning
by Bikram Khanal and Pablo Rivas
Mathematics 2024, 12(9), 1385; https://doi.org/10.3390/math12091385 - 1 May 2024
Cited by 1 | Viewed by 1700
Abstract
Quantum Computing in the Noisy Intermediate-Scale Quantum (NISQ) era has shown promising applications in machine learning, optimization, and cryptography. Despite these progresses, challenges persist due to system noise, errors, and decoherence. These system noises complicate the simulation of quantum systems. The depolarization channel [...] Read more.
Quantum Computing in the Noisy Intermediate-Scale Quantum (NISQ) era has shown promising applications in machine learning, optimization, and cryptography. Despite these progresses, challenges persist due to system noise, errors, and decoherence. These system noises complicate the simulation of quantum systems. The depolarization channel is a standard tool for simulating a quantum system’s noise. However, modeling such noise for practical applications is computationally expensive when we have limited hardware resources, as is the case in the NISQ era. This work proposes a modified representation for a single-qubit depolarization channel. Our modified channel uses two Kraus operators based only on X and Z Pauli matrices. Our approach reduces the computational complexity from six to four matrix multiplications per channel execution. Experiments on a Quantum Machine Learning (QML) model on the Iris dataset across various circuit depths and depolarization rates validate that our approach maintains the model’s accuracy while improving efficiency. This simplified noise model enables more scalable simulations of quantum circuits under depolarization, advancing capabilities in the NISQ era. Full article
(This article belongs to the Special Issue Advances in Quantum Computing and Applications)
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Figure 1

Figure 1
<p>An arbitrary single qubit quantum circuit starting at <math display="inline"><semantics> <mrow> <mo>|</mo> <mn>0</mn> <mo>〉</mo> </mrow> </semantics></math>, applying a Hadamard gate, followed by a sequence of unspecified single quantum gates, then a Pauli-X gate, and finally measurement.</p> Full article ">Figure 2
<p>Scatter plots present the difference between the standard channel and modified depolarization channel expectation value. Each channel was applied to a quantum circuit with single qubit gates of 3, 8, and 15, respectively. The result for 3 single qubit gates is presented in plot (<b>a</b>), while plot (<b>b</b>,<b>c</b>) represent the results for 8 and 15 gates circuits, respectively. The <span class="html-italic">x</span>-axis of each plot represents the number of times the noisy channel was applied and is given by <span class="html-italic">m</span>, while the <span class="html-italic">y</span>-axis gives the varying depolarization rates.</p> Full article ">Figure 3
<p>Feature Mapping of the Iris dataset using Amplitude Encoding and Rotational encoding method. The Rotational encoding scheme, a combination of <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>X</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>Y</mi> </mrow> </semantics></math>, provides better mapping results for the classification problem. The red color represents Class 1, the blue color represents Class 2, and the green color represents Class 3. (<b>a</b>) Bloch Sphere representation of the quantum states obtained by Amplitude encoding of the features vectors. (<b>b</b>) Bloch Sphere representation of the quantum states obtained by Angle encoding of the features vectors.</p> Full article ">Figure 4
<p>Various encoding schemes for single qubits using the rotational encoding. The combination of <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>Z</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>X</mi> </mrow> </semantics></math> gates provides the best mapping for binary classification. The red color represents Class 1, the blue color represents Class 2, and the green color represents Class 3.</p> Full article ">Figure 5
<p>Experimental results for decision boundary evolution presented in the right column and training dynamics in the left column for a QML model on the Iris dataset, with varied noise levels (<span class="html-italic">p</span>) and depolarization channel applied up to (<span class="html-italic">m</span>) times. The decision boundaries are plotted for depths of <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>10</mn> <mo>,</mo> </mrow> </semantics></math> and 15, at noise levels ranging from <math display="inline"><semantics> <mrow> <mn>0.0</mn> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>0.5</mn> <mo>.</mo> </mrow> </semantics></math> The results across rows are presented in chronological order in circuit depth. Accuracy and loss graphs display the model’s performance over 30 epochs, highlighting the impact of noise rate and circuit depth on learning efficacy.</p> Full article ">
18 pages, 487 KiB  
Article
A Formulation of Structural Design Optimization Problems for Quantum Annealing
by Fabian Key and Lukas Freinberger
Mathematics 2024, 12(3), 482; https://doi.org/10.3390/math12030482 - 2 Feb 2024
Cited by 1 | Viewed by 2117
Abstract
We present a novel formulation of structural design optimization problems specifically tailored to be solved by qa. Structural design optimization aims to find the best, i.e., material-efficient yet high-performance, configuration of a structure. To this end, computational optimization strategies can be employed, where [...] Read more.
We present a novel formulation of structural design optimization problems specifically tailored to be solved by qa. Structural design optimization aims to find the best, i.e., material-efficient yet high-performance, configuration of a structure. To this end, computational optimization strategies can be employed, where a recently evolving strategy based on quantum mechanical effects is qa. This approach requires the optimization problem to be present, e.g., as a qubo model. Thus, we develop a novel formulation of the optimization problem. The latter typically involves an analysis model for the component. Here, we use energy minimization principles that govern the behavior of structures under applied loads. This allows us to state the optimization problem as one overall minimization problem. Next, we map this to a qubo problem that can be immediately solved by qa. We validate the proposed approach using a size optimization problem of a compound rod under self-weight loading. To this end, we develop strategies to account for the limitations of currently available hardware. Remarkably, for small-scale problems, our approach showcases functionality on today’s hardware such that this study can lay the groundwork for continued exploration of qa’s impact on engineering design optimization problems. Full article
(This article belongs to the Special Issue Advances in Quantum Computing and Applications)
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Figure 1

Figure 1
<p>A generic elastic body <math display="inline"><semantics> <mo>Ω</mo> </semantics></math> under external loading with prescribed surface traction <math display="inline"><semantics> <msub> <mover accent="true"> <mi>t</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> </semantics></math> and displacement <math display="inline"><semantics> <msub> <mover accent="true"> <mi>u</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> </semantics></math> on the boundary portions <math display="inline"><semantics> <msup> <mo>Γ</mo> <mi>σ</mi> </msup> </semantics></math> and <math display="inline"><semantics> <msup> <mo>Γ</mo> <mi>u</mi> </msup> </semantics></math>, respectively, and body force density <math display="inline"><semantics> <msub> <mi>f</mi> <mi>i</mi> </msub> </semantics></math>.</p> Full article ">Figure 2
<p>Generic setup for a rod under self-weight loading that is composed of multiple elements <span class="html-italic">e</span>.</p> Full article ">Figure 3
<p>Element-wise interpolation functions <math display="inline"><semantics> <mrow> <msubsup> <mi>ϕ</mi> <mi>e</mi> <mrow> <mi mathvariant="normal">I</mi> <mo>/</mo> <mi>II</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and related coefficients <math display="inline"><semantics> <msubsup> <mi>a</mi> <mi>e</mi> <mrow> <mi mathvariant="normal">I</mi> <mo>/</mo> <mi>II</mi> </mrow> </msubsup> </semantics></math> for the approximation of the force functions <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mi>e</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Structural analysis problem: Setup and QUBO pattern. (<b>a</b>) Setup for the composed rod with identical cross sections; (<b>b</b>) pattern of the interactions between the input qubits <math display="inline"><semantics> <msub> <mi>q</mi> <mi>i</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>q</mi> <mi>j</mi> </msub> </semantics></math>.</p> Full article ">Figure 5
<p>Structural analysis problem: solution for the force function <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> obtained by QA compared to the analytical solution <math display="inline"><semantics> <mrow> <msup> <mi>F</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Structural design optimization problem: setup for a composed rod with variable cross sections.</p> Full article ">Figure 7
<p>Structural design optimization problem: QUBO patterns. (<b>a</b>) Pattern of interactions between logical qubits, i.e., <math display="inline"><semantics> <msup> <mi mathvariant="bold-italic">q</mi> <mi>a</mi> </msup> </semantics></math>, <math display="inline"><semantics> <msup> <mi mathvariant="bold-italic">q</mi> <mi>A</mi> </msup> </semantics></math>, and <math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold-italic">q</mi> <mo>^</mo> </mover> </semantics></math>; (b) sub-pattern of interactions between input qubits, i.e., <math display="inline"><semantics> <msup> <mi mathvariant="bold-italic">q</mi> <mi>a</mi> </msup> </semantics></math> and <math display="inline"><semantics> <msup> <mi mathvariant="bold-italic">q</mi> <mi>A</mi> </msup> </semantics></math>.</p> Full article ">Figure 8
<p>Structural design optimization problem: solution for the force function <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> obtained by QA compared to the analytical solution <math display="inline"><semantics> <mrow> <msup> <mi>F</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">
29 pages, 618 KiB  
Article
A Symbolic Approach to Discrete Structural Optimization Using Quantum Annealing
by Kevin Wils and Boyang Chen
Mathematics 2023, 11(16), 3451; https://doi.org/10.3390/math11163451 - 9 Aug 2023
Cited by 6 | Viewed by 1557
Abstract
With the advent of novel quantum computing technologies and the new possibilities thereby offered, a prime opportunity has presented itself to investigate the practical application of quantum computing. This work investigates the feasibility of using quantum annealing for structural optimization. The target problem [...] Read more.
With the advent of novel quantum computing technologies and the new possibilities thereby offered, a prime opportunity has presented itself to investigate the practical application of quantum computing. This work investigates the feasibility of using quantum annealing for structural optimization. The target problem is the discrete truss sizing problem—the goal is to select the best size for each truss member so as to minimize a stress-based objective function. To make the problem compatible with quantum annealing devices, the objective function must be translated into a quadratic unconstrained binary optimization (QUBO) form. This work focuses on exploring the feasibility of making this translation. The practicality of using a quantum annealer for such optimization problems is also assessed. A method is eventually established to translate the objective function into a QUBO form and have it solved by a quantum annealer. However, scaling the method to larger problems faces some challenges that would require further research to address. Full article
(This article belongs to the Special Issue Advances in Quantum Computing and Applications)
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Figure 1

Figure 1
<p>Two-truss system.</p> Full article ">Figure 2
<p>Three-truss system.</p> Full article ">Figure 3
<p>Four-truss system.</p> Full article ">Figure 4
<p>Fractional objective function for the two-truss problem. Global minimum is found to be solution number 7, corresponding to <math display="inline"><semantics><mfenced separators="" open="[" close="]"><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn></mfenced></semantics></math>.</p> Full article ">Figure 5
<p>Fractional objective function for the three-truss problem. Global minimum is found to be solution number 21, corresponding to <math display="inline"><semantics><mfenced separators="" open="[" close="]"><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn></mfenced></semantics></math>.</p> Full article ">Figure 6
<p>Fractional objective function for the four-truss problem. Global minimum is found to be solution number 7, corresponding to <math display="inline"><semantics><mfenced separators="" open="[" close="]"><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn></mfenced></semantics></math>.</p> Full article ">Figure 7
<p>Non-fractional objective function for the two-truss problem. Global minimum is found to be solution number 7, corresponding to <math display="inline"><semantics><mfenced separators="" open="[" close="]"><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn></mfenced></semantics></math>.</p> Full article ">Figure 8
<p>Non-fractional objective function for the three-truss problem. Global minimum is found to be solution number 1, corresponding to <math display="inline"><semantics><mfenced separators="" open="[" close="]"><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn></mfenced></semantics></math>.</p> Full article ">Figure 9
<p>Non-fractional objective function for the four-truss problem. Global minimum is found to be solution number 1, corresponding to <math display="inline"><semantics><mfenced separators="" open="[" close="]"><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn></mfenced></semantics></math>.</p> Full article ">Figure 10
<p>Non-linear scaling function for input coefficients between −1 and 1. The output of the function is shown for various values of the non-linear scaling parameter <math display="inline"><semantics><msub><mi>c</mi><mi mathvariant="italic">NL</mi></msub></semantics></math>.</p> Full article ">Figure 11
<p>Effect of non-linear scaling in two-truss problem for the first iteration of the iterative non-fractional solving method.</p> Full article ">Figure 12
<p>Effect of non-linear scaling in three-truss problem for the first iteration of the iterative non-fractional solving method.</p> Full article ">Figure 13
<p>Effect of non-linear scaling in four-truss problem for the first iteration of the iterative non-fractional solving method.</p> Full article ">Figure 14
<p>Solution probability histogram of two-truss problem using the QA, compared to original objective function. The global optimum solution is located at solution number 7.</p> Full article ">Figure 15
<p>Solution probability histogram of three-truss problem using the QA, compared to original objective function. The global optimum solution is located at solution number 21.</p> Full article ">Figure 16
<p>Solution probability histogram of four-truss problem using the QA, compared to original objective function. The global optimum solution is located at solution number 7.</p> Full article ">
19 pages, 729 KiB  
Article
NISQ-Ready Community Detection Based on Separation-Node Identification
by Jonas Stein, Dominik Ott, Jonas Nüßlein, David Bucher, Mirco Schönfeld and Sebastian Feld
Mathematics 2023, 11(15), 3323; https://doi.org/10.3390/math11153323 - 28 Jul 2023
Cited by 3 | Viewed by 1488
Abstract
The analysis of network structure is essential to many scientific areas ranging from biology to sociology. As the computational task of clustering these networks into partitions, i.e., solving the community detection problem, is generally NP-hard, heuristic solutions are indispensable. The exploration of expedient [...] Read more.
The analysis of network structure is essential to many scientific areas ranging from biology to sociology. As the computational task of clustering these networks into partitions, i.e., solving the community detection problem, is generally NP-hard, heuristic solutions are indispensable. The exploration of expedient heuristics has led to the development of particularly promising approaches in the emerging technology of quantum computing. Motivated by the substantial hardware demands for all established quantum community detection approaches, we introduce a novel QUBO-based approach that only needs number-of-nodes qubits and is represented by a QUBO matrix as sparse as the input graph’s adjacency matrix. The substantial improvement in the sparsity of the QUBO matrix, which is typically very dense in related work, is achieved through the novel concept of separation nodes. Instead of assigning every node to a community directly, this approach relies on the identification of a separation-node set, which, upon its removal from the graph, yields a set of connected components, representing the core components of the communities. Employing a greedy heuristic to assign the nodes from the separation-node sets to the identified community cores, subsequent experimental results yield a proof of concept by achieving an up to 95% optimal solution quality on three established real-world benchmark datasets. This work hence displays a promising approach to NISQ-ready quantum community detection, catalyzing the application of quantum computers for the network structure analysis of large-scale, real-world problem instances. Full article
(This article belongs to the Special Issue Advances in Quantum Computing and Applications)
►▼ Show Figures

Figure 1

Figure 1
<p>Outline of the workflow for the proposed approach of community detection via separation-node identification. The computationally expensive tasks of identifying a set of separation nodes (<b>b</b>) and classifying the communities for these nodes (<b>d</b>) are performed using quantum computing, while the computationally cheap tasks of removing the classified separation-nodes and identifying the resulting connected components (<b>c</b>) are performed classically.</p> Full article ">Figure 2
<p>Counterexample proving no-free-lunch when using Theorem 1 to find surjective separation-node sets.</p> Full article ">Figure 3
<p>Counterexample indicating no-free-lunch when using Theorem 1 to find injective separation-node sets.</p> Full article ">Figure 4
<p>This figure shows the Normalized Mutual Information (NMI) score of the presented approach for 50 different graphs each based on ground truth and a perfect separation edge estimator coupled with the greedy separation-node assignment. The NMI score as defined in [<a href="#B43-mathematics-11-03323" class="html-bibr">43</a>,<a href="#B44-mathematics-11-03323" class="html-bibr">44</a>] was used, as it resembles a well-proven measure for the accuracy of a community given the ground truth [<a href="#B45-mathematics-11-03323" class="html-bibr">45</a>]. The different probabilities for intra-community edges in the chosen SBM model resemble different difficulties according to the phase transition known for this model. The lower the stated probability, the harder the problem. The probabilities were chosen such that the hardest graphs barely differed from a null model inheriting no measurable structure up to the hardest that still allowed perfect NMI scores. For this dataset, the phase transition can be calculated to be at a probability of 0.2865 for the intra-community edges. As modularity maximization has been shown to perform very well up until the sharp phase transition (which is not reached here), the constantly good results for the SA based approach appear to be reasonable. Additional tests show a sharp performance drop off to NMI values at around 0.5 for smaller intra-probabilities such as 0.23.</p> Full article ">Figure 5
<p>This box plot displays the fraction of the achieved modularity score by the best known solution for selected standard benchmark datasets: (1) the social network of a karate club [<a href="#B46-mathematics-11-03323" class="html-bibr">46</a>], (2) the social interactions between dolphins [<a href="#B47-mathematics-11-03323" class="html-bibr">47</a>], (3) the collectively appearing characters in the book “Les Miserables” [<a href="#B48-mathematics-11-03323" class="html-bibr">48</a>], (4) protein–protein interactions [<a href="#B49-mathematics-11-03323" class="html-bibr">49</a>] and (5) jointly bought political books [<a href="#B50-mathematics-11-03323" class="html-bibr">50</a>]. Each graph was analyzed 10 times using simulated annealing. Our approach clearly does not work well for the karate club network. Closer inspections yield the result signifying that the connected components resulting from the found separation-node sets often only consist of single nodes, indicating suboptimality in using neighborhood connectivity for this relatively small dataset.</p> Full article ">Figure 6
<p>The <span class="html-italic">y</span>-axis depicts the deviation factor from the best-known separation-node set in size. Notably, the absolute sizes of the identified separation-node sets are typically similar over the different difficulties, while they rise slightly for larger graphs.</p> Full article ">Figure 7
<p><math display="inline"><semantics><msup><mi>R</mi><mn>2</mn></msup></semantics></math> score of the edge-neighborhood-connectivity-based separation edge estimator. In practice, an <math display="inline"><semantics><msup><mi>R</mi><mn>2</mn></msup></semantics></math> score of 30% implies that merely 30% of the variability of the ground truth has been accounted for. A strict trend towards worse results for harder datasets is clearly visible. This shows that the performance of the estimator decreases for harder problem instances as to be expected while still yielding somewhat accurate results.</p> Full article ">Figure 8
<p>This figure depicts the normalized mutual information score of the selected SBM benchmark graphs using the greedy assignment of separation nodes to communities. A substantial drop off in performance can be observed for the harder datasets. Meanwhile, as all problem instances are significantly above the phase transition for modularity maximization in these datasets (an intra-prob of 0.2865), our classical baseline easily identifies close to optimal solutions. Notably, however, it is promisingly slightly outperformed by our approach in the case of the easiest dataset.</p> Full article ">Figure 9
<p>This figure depicts the normalized mutual information score of the selected SBM benchmark graph using a simulated annealing-based approach of assigning the separation nodes to communities. The worse performance for the easy dataset clearly indicates that the chosen simulated annealing approach based on the QUBO as described in <a href="#sec3dot6-mathematics-11-03323" class="html-sec">Section 3.6</a> is suboptimal in general.</p> Full article ">
18 pages, 841 KiB  
Article
Finding Debt Cycles: QUBO Formulations for the Maximum Weighted Cycle Problem Solved Using Quantum Annealing
by Hendrik Künnemann and Frank Phillipson
Mathematics 2023, 11(12), 2741; https://doi.org/10.3390/math11122741 - 16 Jun 2023
Cited by 1 | Viewed by 2003
Abstract
The problem of finding the maximum weighted cycle in a directed graph map to solve optimization problems is NP-hard, implying that approaches in classical computing are inefficient. Here, Quantum computing might be a promising alternative. Many current approaches to the quantum computer [...] Read more.
The problem of finding the maximum weighted cycle in a directed graph map to solve optimization problems is NP-hard, implying that approaches in classical computing are inefficient. Here, Quantum computing might be a promising alternative. Many current approaches to the quantum computer are based on a Quadratic Unconstrained Binary Optimization (QUBO) problem formulation. This paper develops four different QUBO approaches to this problem. The first two take the starting vertex and the number of vertices used in the cycle as given, while the latter two loosen the second assumption of knowing the size of the cycle. A QUBO formulation is derived for each approach. Further, the number of binary variables required to encode the maximum weighted cycle problem with one or both assumptions for the respective approach is made explicit. The problem is motivated by finding the maximum weighted debt cycle in a debt graph. This paper compares classical computing versus currently available (hybrid) quantum computing approaches for various debt graphs. For the classical part, it investigated the Depth-First-Search (DFS) method and Simulated Annealing. For the (hybrid) quantum approaches, a direct embedding on the quantum annealer and two types of quantum hybrid solvers were utilized. Simulated Annealing and the usage of the hybrid CQM (Constrained Quadratic Model) had promising functionality. The DFS method, direct QPU, and hybrid BQM (Binary Quadratic Model), on the other hand, performed less due to memory issues, surpassing the limit of decision variables and finding the right penalty values, respectively. Full article
(This article belongs to the Special Issue Advances in Quantum Computing and Applications)
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<p>Depth-First-Search. (<b>a</b>) A random directed graph. (<b>b</b>) Respective DFS tree.</p> Full article ">Figure 2
<p>Simple cycle using 4 vertices, with Node 5 being a duplicate of Node 1.</p> Full article ">Figure 3
<p>Huangdao Zone graph without vertices of in-/out-degree 0.</p> Full article ">
14 pages, 739 KiB  
Article
A Depth-Progressive Initialization Strategy for Quantum Approximate Optimization Algorithm
by Xinwei Lee, Ningyi Xie, Dongsheng Cai, Yoshiyuki Saito and Nobuyoshi Asai
Mathematics 2023, 11(9), 2176; https://doi.org/10.3390/math11092176 - 5 May 2023
Cited by 7 | Viewed by 2053
Abstract
The quantum approximate optimization algorithm (QAOA) is known for its capability and universality in solving combinatorial optimization problems on near-term quantum devices. The results yielded by QAOA depend strongly on its initial variational parameters. Hence, parameter selection for QAOA becomes an active area [...] Read more.
The quantum approximate optimization algorithm (QAOA) is known for its capability and universality in solving combinatorial optimization problems on near-term quantum devices. The results yielded by QAOA depend strongly on its initial variational parameters. Hence, parameter selection for QAOA becomes an active area of research, as bad initialization might deteriorate the quality of the results, especially at great circuit depths. We first discuss the patterns of optimal parameters in QAOA in two directions: the angle index and the circuit depth. Then, we discuss the symmetries and periodicity of the expectation that is used to determine the bounds of the search space. Based on the patterns in optimal parameters and the bounds restriction, we propose a strategy that predicts the new initial parameters by taking the difference between the previous optimal parameters. Unlike most other strategies, the strategy we propose does not require multiple trials to ensure success. It only requires one prediction when progressing to the next depth. We compare this strategy with our previously proposed strategy and the layerwise strategy for solving the Max-cut problem in terms of the approximation ratio and the optimization cost. We also address the non-optimality in previous parameters, which is seldom discussed in other works despite its importance in explaining the behavior of variational quantum algorithms. Full article
(This article belongs to the Special Issue Advances in Quantum Computing and Applications)
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<p>Optimal parameters variation of a 10-node Erdös-Rényi graph with edge probability of 0.7. (<b>a</b>) The variation of the optimal parameters at fixed circuit depth <span class="html-italic">p</span> against the angle index <span class="html-italic">j</span>. It shows the adiabatic path of the parameters with increasing <math display="inline"><semantics> <mi>γ</mi> </semantics></math> and decreasing <math display="inline"><semantics> <mi>β</mi> </semantics></math>. (<b>b</b>) The variation of the optimal parameters at fixed angle index <span class="html-italic">j</span> against the circuit depth <span class="html-italic">p</span>. It shows the non-optimality of the parameters with decreasing <math display="inline"><semantics> <mi>γ</mi> </semantics></math> and increasing <math display="inline"><semantics> <mi>β</mi> </semantics></math>. (<b>c</b>) The landscape of <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> normalized expectation (i.e., <math display="inline"><semantics> <mi>α</mi> </semantics></math>) against <math display="inline"><semantics> <msub> <mi>γ</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>β</mi> <mn>1</mn> </msub> </semantics></math>. The symmetry is shown by the <math display="inline"><semantics> <mrow> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>=</mo> <mi>π</mi> </mrow> </semantics></math> axis and the periodicity is shown by the <math display="inline"><semantics> <mrow> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math> axis. It can be seen that the landscape in <math display="inline"><semantics> <mrow> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>∈</mo> <mrow> <mo>[</mo> <mi>π</mi> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> is the landscape in <math display="inline"><semantics> <mrow> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>∈</mo> <mrow> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> rotated by <math display="inline"><semantics> <msup> <mn>180</mn> <mo>∘</mo> </msup> </semantics></math>, and the landscape just repeats itself beyond <math display="inline"><semantics> <mrow> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>Visualization of the bilinear strategy. <math display="inline"><semantics> <msub> <mo>Δ</mo> <mn>1</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mo>Δ</mo> <mn>2</mn> </msub> </semantics></math>, and <math display="inline"><semantics> <msub> <mo>Δ</mo> <mn>3</mn> </msub> </semantics></math> correspond to the values calculated in Equations (<a href="#FD11-mathematics-11-02176" class="html-disp-formula">11</a>), (<a href="#FD12-mathematics-11-02176" class="html-disp-formula">12</a>), and (<a href="#FD13-mathematics-11-02176" class="html-disp-formula">13</a>) respectively. <math display="inline"><semantics> <msub> <mo>Δ</mo> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mo>Δ</mo> <mn>2</mn> </msub> </semantics></math> represent the change due to the non-optimality. <math display="inline"><semantics> <msub> <mo>Δ</mo> <mn>3</mn> </msub> </semantics></math> represents the change due to the adiabatic path. <math display="inline"><semantics> <msub> <mo>Φ</mo> <mi>p</mi> </msub> </semantics></math> is the new initial parameters extrapolated from <math display="inline"><semantics> <msubsup> <mo>Φ</mo> <mrow> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>*</mo> </msubsup> </semantics></math> and <math display="inline"><semantics> <msubsup> <mo>Φ</mo> <mrow> <mi>p</mi> <mo>−</mo> <mn>2</mn> </mrow> <mo>*</mo> </msubsup> </semantics></math>.</p> Full article ">Figure 3
<p>Comparison of the results for parameters fixing, layerwise, and the newly proposed bilinear strategy, for 4 graph instances extracted from the 30 graph instances evaluated. <span class="html-italic">n</span> is the number of nodes/vertices of the graph, <span class="html-italic">d</span> is the degree for regular graphs, ‘prob’ is the edge probability for Erdös-Rényi graphs. (<b>a</b>–<b>d</b>) show the changes in the approximation ratio <math display="inline"><semantics> <mi>α</mi> </semantics></math> against <span class="html-italic">p</span>. (<b>e</b>–<b>h</b>) show the <math display="inline"><semantics> <msub> <mi>n</mi> <mi>fev</mi> </msub> </semantics></math> required before convergence at different <span class="html-italic">p</span>’s for the L-BFGS-B optimizer (log scale). For parameters fixing and layerwise, the <math display="inline"><semantics> <msub> <mi>n</mi> <mi>fev</mi> </msub> </semantics></math> is the total of 20 trials.</p> Full article ">Figure 4
<p>(<b>a</b>) <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> normalized expectation (i.e., <math display="inline"><semantics> <mi>α</mi> </semantics></math>) landscape for a 10-node 3-regular graph showing multiple maxima in <math display="inline"><semantics> <mrow> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>∈</mo> <mrow> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. When used as a starting point in the bilinear strategy, the maximum on the left follows the adiabatic path, whereas the maximum on the right does not follow the adiabatic path. (<b>b</b>) The variation of the optimal parameters with a non-adiabatic start. Unlike the adiabatic start, the <math display="inline"><semantics> <mi>β</mi> </semantics></math>’s oscillate back and forth. (<b>c</b>) The effect of the bilinear strategy under the non-adiabatic start.</p> Full article ">
15 pages, 1969 KiB  
Article
Graph Generation for Quantum States Using Qiskit and Its Application for Quantum Neural Networks
by Alexandru-Gabriel Tudorache
Mathematics 2023, 11(6), 1484; https://doi.org/10.3390/math11061484 - 18 Mar 2023
Cited by 1 | Viewed by 2761
Abstract
This paper describes a practical approach to the quantum theory using the simulation and processing technology available today. The proposed project allows us to create an exploration graph so that for an initial starting configuration of the qubits, all possible states are created [...] Read more.
This paper describes a practical approach to the quantum theory using the simulation and processing technology available today. The proposed project allows us to create an exploration graph so that for an initial starting configuration of the qubits, all possible states are created given a set of gates selected by the user. For each node in the graph, we can obtain various types of information such as the applied gates from the initial state (the transition route), necessary cost, representation of the quantum circuit, as well as the amplitudes of each state. The project is designed not as an end goal, but rather as a processing platform that allows users to visualize and explore diverse solutions for different quantum problems in a much easier manner. We then describe some potential applications of this project in other research fields, illustrating the way in which the states from the graph can be used as nodes in a new interpretation of a quantum neural network; the steps of a hybrid processing chain are presented for the problem of finding one or more states that verify certain conditions. These concepts can also be used in academia, with their implementation being possible with the help of the Python programming language, the NumPy library, and Qiskit—the open-source quantum framework developed by IBM. Full article
(This article belongs to the Special Issue Advances in Quantum Computing and Applications)
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<p>Simple entanglement circuit.</p> Full article ">Figure 2
<p>UML Class diagram of the project.</p> Full article ">Figure 3
<p>The graph for the presented example, with quantum states as nodes, for a one-qubit register and three selected gates (X, Z, and H).</p> Full article ">Figure 4
<p>Design of a neural network using the quantum states from the graph. The configuration of the output layer depends on the states from the inner layers; the data from the graph, analyzed at different exploration depths, were used to populate the final nodes in each neural network architecture.</p> Full article ">Figure 5
<p>The scheme for the hybrid classical–quantum solution, integrating the main processing blocks (the neural network) with a quantum oracle.</p> Full article ">
27 pages, 5376 KiB  
Article
Heart Failure Detection Using Instance Quantum Circuit Approach and Traditional Predictive Analysis
by Shtwai Alsubai, Abdullah Alqahtani, Adel Binbusayyis, Mohemmed Sha, Abdu Gumaei and Shuihua Wang
Mathematics 2023, 11(6), 1467; https://doi.org/10.3390/math11061467 - 17 Mar 2023
Cited by 10 | Viewed by 2547
Abstract
The earlier prediction of heart diseases and appropriate treatment are important for preventing cardiac failure complications and reducing the mortality rate. The traditional prediction and classification approaches have resulted in a minimum rate of prediction accuracy and hence to overcome the pitfalls in [...] Read more.
The earlier prediction of heart diseases and appropriate treatment are important for preventing cardiac failure complications and reducing the mortality rate. The traditional prediction and classification approaches have resulted in a minimum rate of prediction accuracy and hence to overcome the pitfalls in existing systems, the present research is aimed to perform the prediction of heart diseases with quantum learning. When quantum learning is employed in ML (Machine Learning) and DL (Deep Learning) algorithms, complex data can be performed efficiently with less time and a higher accuracy rate. Moreover, the proposed ML and DL algorithms possess the ability to adapt to predictions with alterations in the dataset integrated with quantum computing that provides robustness in the earlier detection of chronic diseases. The Cleveland heart disease dataset is being pre-processed for the checking of missing values to avoid incorrect predictions and also for improvising the rate of accuracy. Further, SVM (Support Vector Machine), DT (Decision Tree) and RF (Random Forest) are used to perform classification. Finally, disease prediction is performed with the proposed instance-based quantum ML and DL method in which the number of qubits is computed with respect to features and optimized with instance-based learning. Additionally, a comparative assessment is provided for quantifying the differences between the standard classification algorithms with quantum-based learning in order to determine the significance of quantum-based detection in heart failure. From the results, the accuracy of the proposed system using instance-based quantum DL and instance-based quantum ML is found to be 98% and 83.6% respectively. Full article
(This article belongs to the Special Issue Advances in Quantum Computing and Applications)
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<p>The overall architecture of the proposed system.</p> Full article ">Figure 2
<p>Architectural framework of the proposed classification algorithm.</p> Full article ">Figure 3
<p>Block Diagram of Quantum-based ML algorithm.</p> Full article ">Figure 4
<p>QDL Functional Diagram.</p> Full article ">Figure 5
<p>Correlation matrix representation.</p> Full article ">Figure 6
<p>Classification of patients with heart disease.</p> Full article ">Figure 7
<p>Confusion matrix for SVC.</p> Full article ">Figure 8
<p>ROC graph for SVC.</p> Full article ">Figure 9
<p>DT evaluation with the confusion matrix.</p> Full article ">Figure 10
<p>ROC estimation for DT Classifier.</p> Full article ">Figure 11
<p>RF assessment with the confusion matrix.</p> Full article ">Figure 12
<p>ROC analysis for RF.</p> Full article ">Figure 13
<p>Proposed confusion matrix representation.</p> Full article ">Figure 14
<p>ROC analysis for proposed quantum approach.</p> Full article ">Figure 15
<p>Graphical performance analysis of proposed QML.</p> Full article ">Figure 16
<p>Graphical Representation of the Proposed QDL.</p> Full article ">

Review

Jump to: Research

18 pages, 303 KiB  
Review
Quantum Computing in Telecommunication—A Survey
by Frank Phillipson
Mathematics 2023, 11(15), 3423; https://doi.org/10.3390/math11153423 - 6 Aug 2023
Cited by 9 | Viewed by 4945
Abstract
Quantum computing, an emerging paradigm based on the principles of quantum mechanics, has the potential to revolutionise various industries, including Telecommunications. This paper explores the transformative impact of quantum computing on the telecommunication market, focusing on its applications in solving computationally intensive problems. [...] Read more.
Quantum computing, an emerging paradigm based on the principles of quantum mechanics, has the potential to revolutionise various industries, including Telecommunications. This paper explores the transformative impact of quantum computing on the telecommunication market, focusing on its applications in solving computationally intensive problems. By leveraging the inherent properties of quantum systems, such as superposition and entanglement, quantum computers offer the promise of exponential computational speedup and enhanced problem-solving capabilities. This paper provides an in-depth analysis of the current state of quantum computing in telecommunication, examining key algorithms and approaches, discussing potential use cases, and highlighting the challenges and future prospects of this disruptive technology. Full article
(This article belongs to the Special Issue Advances in Quantum Computing and Applications)
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<p>Amplitude encoding: two-qubit circuit for data encoding of two normalised two-dimensional data points.</p> Full article ">
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