Quantum Error Correction: Symmetric, Asymmetric, Synchronizable, and Convolutional Codes

234 Pages·2020·2.969 MB·English

by  Giuliano Gadioli La Guardia

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.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor
for any errors or omissions that may have been made. The publisher remains neutral with regard to
jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
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
[e-mail]
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