Quantum Science and Technology Giuliano Gadioli La Guardia Quantum Error Correction Symmetric, Asymmetric, Synchronizable, and Convolutional Codes Quantum Science and Technology Series Editors Raymond Laflamme, Waterloo, ON, Canada Gaby Lenhart, Sophia Antipolis, France Daniel Lidar, Los Angeles, CA, USA Arno Rauschenbeutel, Vienna University of Technology, Vienna, Austria Renato Renner, Institut für Theoretische Physik, ETH Zürich, Zürich, Switzerland MaximilianSchlosshauer,DepartmentofPhysics,UniversityofPortland,Portland, OR, USA Jingbo Wang, Department of Physics, University of Western Australia, Crawley, WA, Australia Yaakov S. Weinstein, Quantum Information Science Group, The MITRE Corporation, Princeton, NJ, USA H. M. Wiseman, Brisbane, QLD, Australia The book series Quantum Science and Technology is dedicated to one of today’s mostactiveandrapidlyexpandingfieldsofresearchanddevelopment.Inparticular, the series will be a showcase for the growing number of experimental implemen- tations and practical applications of quantum systems. These will include, but are not restricted to: quantum information processing, quantum computing, and quantum simulation; quantum communication and quantum cryptography; entan- glement and other quantum resources; quantum interfaces and hybrid quantum systems; quantum memories and quantum repeaters; measurement-based quantum control and quantum feedback; quantum nanomechanics, quantum optomechanics and quantum transducers; quantum sensing and quantum metrology; as well as quantum effects in biology. Last but not least, the series will include books on the theoretical and mathematical questions relevant to designing and understanding these systems and devices, as well as foundational issues concerning the quantum phenomena themselves. Written and edited by leading experts, the treatments will be designed for graduate students and other researchers already working in, or intending to enter the field of quantum science and technology. More information about this series at http://www.springer.com/series/10039 Giuliano Gadioli La Guardia Quantum Error Correction Symmetric, Asymmetric, Synchronizable, and Convolutional Codes 123 Giuliano Gadioli La Guardia Department ofMathematics andStatistics PontaGrossa State University PontaGrossa, Paraná,Brazil ISSN 2364-9054 ISSN 2364-9062 (electronic) QuantumScience andTechnology ISBN978-3-030-48550-4 ISBN978-3-030-48551-1 (eBook) https://doi.org/10.1007/978-3-030-48551-1 ©SpringerNatureSwitzerlandAG2020 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland To my parents and my family Preface Theoryofquantuminformationandcomputationhasbeenextensivelyinvestigated in the last two decades (see the textbook [121] by Nielsen and Chuang and the references [25, 53, 144, 147, 148] for the first constructions or construction methodsofquantumcodesshownintheliterature;seealsothepapers[1,2,20,21, 82, 122, 146] concerned with constructions of topological quantum codes, which will not be investigated in this book). This book is written in order to familiarize graduate or postgraduate students with respect to several constructions or con- struction methods of quantum codes as well as techniques of constructions of quantum convolutional codes available in the literature. More precisely, we gath- ered great part of the most relevant papers that we have published concerning quantum coding theory to present such results here, in form of book. To this end, we utilize the well-known Calderbank–Shor–Steane (CSS) construction, the Hermitian and the Steane enlargement construction to certain classes of classical codes.Thesequantumcodeshavegoodparametersandtheyareintroducedrecently in the literature. Furthermore, the book also presents several constructions of families of asymmetric quantum codes with good parameters. Keepinginmindasimilarapproach,thebookalsocontainsacarefuldescription oftheproceduresadoptedtoconstructfamiliesofquantumconvolutionalcodes.To close the book, we introduce and construct families of asymmetric quantum con- volutional codes (this concept was introduced in Ref. [102] (La Guardia, G.G.: Asymmetric quantum convolutional codes. Quantum Inform. Processing 15, 167– 183 (2016)). Although the book does not bring new didactic approach nor new form of presentation, I tried to write it carefully, with accessible language and clear explanations,inordertoimprovethequalityandtheaccessibilityofit.Inmypoint of view, the book has some advantages. The first one is to teach the reader certain algebraic techniques of code construction that could improve the capacity of abstraction of him/her. It is also an attempt to motivate the reader to perform their owncontributionsfromthisareaofresearch.Anotherimportantcontributionisthat vii viii Preface all constructions presented here are performed algebraically, i.e., the procedures adopted are capable of constructing families of codes, and not only codes with specific parameters. Description of the Book The book is organized in such a way that the reader can skip some introductory chapters without major problems. Chapter1presentsareviewofsomebasicconceptsonlinearalgebraandmetric spaces,necessarytodefinethescenarioandthestructureofthequantummechanics. In Chap. 2, the postulates of quantum mechanics, the definition of single and multiplequbitgatesandthemostcommontypesofquantumchannelsarereviewed. Chapter 3 is concerned with the first constructions or construction methods of quantum codes shown in the literature. The well-known five qubit and the Steane code are examples of such codes. We also review the CSS construction and the stabilizer quantum code construction. Chapter 4 is devoted to review some definitions and results on linear block codes. We recall the concept of Euclidean and Hermitian dual of a linear code as well as the techniques to obtain new codes from old. Additionally, the classes of cyclicandalgebraicgeometrycodesarereviewed,sincesuchclassicallinearcodes are necessary for the quantum code constructions presented in this book. Chapter5bringsthemostrelevantconstructionsofquantumcodesthatwehave publishedinthelasttenyearsofresearch.Theyincludeseveralfamiliesofquantum codes derived from (classical) Bose–Chaudhuri–Hocquenghem (BCH) and from (classical) algebraic geometry codes. Moreover, constructions of quantum syn- chronizable codesderived from (classical) cyclic, BCH and product codes arealso presented here. InChap.6,asinChap.5,wepresentmymostrelevantcontributionsconcerning asymmetric quantum code constructions, which were published in the last years. We construct several families of asymmetric quantum codes (AQQs) derived from (classical) Reed–Solomon and generalized Reed–Solomon codes, generalized Reed–Muller and BCH codes. Additionally, we generalize to AQQs the well-known methods which are valid to quantum codes, namely: puncturing, extending, expanding, direct sum and the ðujuþvÞ construction. In Chap. 7, we present my main contributions published in the last years con- cerning constructions of quantum convolutional codes. We explain how to con- structfamiliesofquantumconvolutionalcodeswithgoodparametersderivedfrom Preface ix (classical) convolutional codes. These classical convolutional codes were obtained fromlinearblockcodes:BCH,negacyclicandalgebraicgeometrycodes.Moreover, we introduce a new class of codes: the asymmetric quantum convolutional codes. Have a good read and enjoy the book!! Ponta Grossa, Paraná, Brazil Giuliano Gadioli La Guardia [email protected] Contents 1 Some Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Linear Transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Diagonalizable Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.6 Metric Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.7 Inner Product Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.8 Commutator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 A Little Bit of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1 Postulates of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Ensemble of Quantum States. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Universal Quantum Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.1 Single Qubit Gates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.2 Multiple Qubit Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.3 Quantum Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3 Quantum Error-Correcting Codes . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 The Shor Code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2.1 Three Qubit Bit Flip Code . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2.2 Three Qubit Phase Flip Code . . . . . . . . . . . . . . . . . . . . . . 28 3.2.3 The Shor Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3 The Steane Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4 Five-Qubit Code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.5 Stabilizer Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.6 Calderbank–Shor–Steane Construction. . . . . . . . . . . . . . . . . . . . . 38 xi