Quantum Image Processing Fei Yan • Salvador E. Venegas-Andraca

Quantum Image Processing Fei Yan Salvador E. Venegas-Andraca Changchun University of Science and Tecnologico de Monterrey Technology Escuela de Ingenieria y Ciencias Changchun, China Monterrey, Mexico

ISBN 978-981-32-9330-4 ISBN 978-981-32-9331-1 (eBook) https://doi.org/10.1007/978-981-32-9331-1

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This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Preface

Quantum information science is a deeply interdisciplinary field that involves physics, mathematics, and computer science. It is devoted to finding methods to exploit quantum mechanical effects in nature, notably superposition and entan- glement, to perform information processing beyond the limits of conventional computation. Over the past several decades, various research groups around the world have strived to achieve the ambitious goal of building a quantum computer that could dramatically improve computational power for particular tasks. Responding to the growing need to extract information from images and video, image processing is a fundamental task in many branches of science and engineering. Due to the restricted architecture of classical computers and the com- putational complexity of state-of-the-art classical algorithms in image processing and its applications, developing efficient algorithms to store and manipulate visual information has become an important and challenging research area. Quantum image processing focuses on quantum algorithms for storing, process- ing, and retrieving visual information. Due to some of the astounding properties inherent to quantum information, for instance, computational parallelism, it is anticipated that quantum image processing technologies will offer capabilities and performance that are currently unrivaled by their traditional equivalents in areas such as computing speed, tamperproof security, and minimal storage requirements. This book is divided into seven chapters. In Chap. 1, the key fundamentals of quantum computation and information are reviewed, and the history of quantum image processing is introduced. The widely used quantum image representations and their well-designed operations are presented in Chaps. 2 and 3. The outline of quantum image security technologies and a few quantum image understanding algorithms are suggested in Chaps. 4 and 5. The two emerging subtopics of quantum movies and quantum audio are elaborated in Chap. 6. Chapter 7 discusses open questions identified in the literature, along with future development trends in quantum image processing. It is hoped that this book offers a rigorous introduction to quantum image processing and some thought-provoking snapshots of prospective developments. The completion of this book relied greatly on the research achievements published

v vi Preface in the field and the two bibles: Quantum Computation and Quantum Information (Michael A. Nielsen et al.) and Digital Image Processing (Rafael C. Gonzalez et al.). Immense gratitude is due to the emeritus professor of the Tokyo Institute of Technology, Kaoru Hirota, and Professor Zhengang Jiang at Changchun University of Science and Technology for their enlightenment and ongoing help that turned this book from an idea into reality. In addition, special commendation goes to Kehan Chen, Nianqiao Li, and Shan Zhao for their contributions to the timely and thorough organization of the figures and references in the book. This work is supported by the National Natural Science Foundation of China (No. 61502053). SEVAgratefully acknowledges the financial support of CONACyT (SNI 41594) and Fronteras de la Ciencia (1007). Additionally, SEVA dedicates his work to his dearest wife Lourdes and beloved daughter Renata, his eternal gratitude for their love, support, and patience. Being subject to the limits of the authors’ ability and because quantum image processing is still in its primary stage, it is hard to avoid errors and omissions. The authors apologize for this and welcome criticism and suggestions.

Changchun, China Fei Yan Monterrey, Mexico Salvador E. Venegas-Andraca Contents

1 Introduction and Overview ...... 1 1.1 QuantumComputationandInformation...... 2 1.1.1 QuantumComputers...... 2 1.1.2 QuantumBitsandQuantumRegisters...... 4 1.1.3 QuantumCircuitsandQuantumGates ...... 7 1.2 Background of Quantum Image Processing ...... 9 1.2.1 QuantumInterdisciplinaryResearch...... 9 1.2.2 Quantum-BasedImageProcessingTechniques ...... 11 References...... 15 2 Quantum Image Representations ...... 19 2.1 QuantumImageModels ...... 19 2.1.1 ImageColorModel...... 19 2.1.2 ImageCoordinateModel...... 21 2.1.3 ImageColorInformationEncodingModel...... 22 2.2 FlexibleRepresentationforQuantumImages...... 23 2.2.1 FRQI Representation and Initialization...... 23 2.2.2 GeometricTransformationsonFRQIImages...... 26 2.2.3 ColorTransformationsonFRQIImages ...... 31 2.3 Multi-Channel Representation for Quantum Images ...... 34 2.3.1 RGBα ColorSpace...... 34 2.3.2 MCQI Representation and Initialization ...... 35 2.3.3 ColorTransformationsonMCQIImages...... 37 2.4 NovelEnhancedRepresentationforQuantumImages...... 42 2.4.1 NEQR Representation and Initialization ...... 42 2.4.2 ColorTransformationsonNEQRImages ...... 43 References...... 47 3 Quantum Image Operations ...... 49 3.1 Parallel Comparison of Multiple Pairs of Quantum Images ...... 49 3.1.1 Representation of Strip Using Multiple Quantum Images .... 50 3.1.2 SchemetoCompareQuantumImagesinParallel...... 51

vii viii Contents

3.1.3 EvaluationofSimilarityBetweenTwoQuantumImages..... 53 3.1.4 Parallel Comparison of Multiple Quantum Images ina Strip...... 55 3.2 Quantum Image Up-Scaling Based on Nearest-Neighbor Interpolation...... 59 3.2.1 GeneralizedQuantumImageRepresentation...... 59 3.2.2 SchemeofQuantumImageUp-ScalingOperation ...... 60 3.2.3 CircuitImplementationofUp-ScalingOperation...... 61 3.2.4 ExampleofQuantumImageUp-Scaling...... 63 3.3 QuantumImageRotationbyanArbitraryAngle...... 65 3.3.1 ThreeShearTransformations ...... 65 3.3.2 Quantum Modules for Shear Transformations ...... 66 3.3.3 SchemeofQuantumImageRotationOperation ...... 70 3.3.4 ExampleofQuantumImageRotation ...... 74 References...... 77 4 Quantum Image Security ...... 79 4.1 QIMP-Based Security Technologies ...... 79 4.1.1 AlgorithmsforQuantumWatermarking...... 80 4.1.2 AlgorithmsforQuantumImageEncryption...... 82 4.1.3 Algorithms for Quantum Image Steganography ...... 84 4.2 DupleWatermarkingStrategyforQuantumImages ...... 85 4.2.1 Double Information Key Generation...... 85 4.2.2 WatermarkImageEmbeddingandExtraction...... 90 4.2.3 Metric for Estimating Congruity Between Quantum Images...... 92 4.3 Quantum Image Encryption Using One-Dimensional Quantum CellularAutomata...... 93 4.3.1 QuantumCellularAutomata ...... 93 4.3.2 SchemeofQuantumImageEncryptionOperation...... 94 4.3.3 ExampleofQuantumImageEncryption ...... 96 4.4 LSB-Based Quantum Image Steganography Algorithm ...... 96 4.4.1 Plain LSB Steganography Operation ...... 97 4.4.2 Block LSB Steganography Operation ...... 98 4.4.3 Example of Quantum Image Steganography ...... 100 References...... 102 5 Quantum Image Understanding ...... 105 5.1 LocalFeaturePointExtractionforQuantumImages ...... 105 5.1.1 QuantumImageColorTransformations...... 106 5.1.2 Quantum Image Addition and Subtraction Operations ...... 107 5.1.3 SchemeofQuantumImageFeatureExtraction ...... 108 5.2 Quantum Image Median Filtering in Spatial Domain ...... 110 5.2.1 Quantum Modules for Median Filter ...... 111 5.2.2 Circuit Implementation of Median Filtering Operation ...... 112 5.2.3 Example of Quantum Image Filtering ...... 116 Contents ix

5.3 Threshold-BasedQuantumImageSegmentation...... 117 5.3.1 CQIR Representation and Initialization ...... 117 5.3.2 CircuitImplementationofSegmentationOperation ...... 118 5.3.3 ExampleofQuantumImageSegmentation ...... 120 References...... 122 6 Quantum Multimedia Techniques ...... 125 6.1 ChromaticFrameworkforQuantumMoviesandApplications...... 125 6.1.1 GrayscaleQuantumMovieScheme ...... 125 6.1.2 ChromaticFrameworkforQuantumMovies...... 127 6.1.3 QuantumOperationstoAchieveMontages ...... 132 6.2 Flexible Representation and Manipulation of Quantum Audio Signals...... 138 6.2.1 QuantumRepresentationforDigitalAudio ...... 138 6.2.2 FlexibleRepresentationofQuantumAudioSignals...... 140 6.2.3 QuantumOperationstoManipulateAudioSignals...... 146 References...... 156 7 Summary and Discussion ...... 159 7.1 ConcludingRemarks...... 159 7.2 OpenQuestionsandFutureDirections...... 160 References...... 163 A Appendix ...... 165 References...... 171