Holdings: Quantum error correction and fault tolerant quantum computing

Quantum error correction and fault tolerant quantum computing:

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Bibliographic Details
Main Author:	Gaitan, Frank (Author)
Format:	Book
Language:	English
Published:	Boca Raton, Fla. [u.a.] CRC Press 2008
Subjects:	Quantum computers Error-correcting codes (Information theory) Fault-tolerant computing Fehlerkorrekturcode Fehlertoleranz Quantencomputer
Online Access:	Inhaltsverzeichnis
Item Description:	Includes bibliographical references and index
Physical Description:	XVIII, 292 S. graph. Darst.
ISBN:	9780849371998

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MARC


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adam_text	CONTENTS LIST OF FIGURES XI* LIST OF TABLES XV* PREFACE XVII* 1 INTRODUETION 1* 1.1 HISTORICAL BACKGROUND 2* 1.2 CLASSICAL ERROR CORRECTING CODES 4* 1.2.1 LINEAR ERROR CORRECTING CODES 4* 1.2.2 ERRORS, HAMMING WEIGHT AND DISTANCE . 7* 1.2.3 ERROR DETECTION AND CORRECTION 10* 1.2.4 ERRORPROBABILITY . . . . . . . . . . . . . 12* 1.2.5 BOUNDS ON CODE PARAMETERS. . . . . . . 14* 1.3 USING QUANTUM SYSTEMS TO STORE AND PROCESS DATA 17* 1.3.1 LINEAR SUPERPOSITION AND QUANTUM PARALLELISM 17* 1.3.2 NO-CLONING THEOREM . . . . . . 20* 1.3.3 MEASUREMENT.......... 21* 1.3.4 DISTINGUISHING QUANTUM STATES 23* 1.3.5 ENTANGLEMENT.......... 24* 1.3.6 NOISE: ERRORS AND DECOHERENCE 28* 1.4 QUANTUM ERROR CORRECTING CODES-FIRST PASS 31* 1.4.1 REDUNDANCY WITHOUT CLONING . . . . 31* 1.4.2 NECESSARY AND SUFFICIENT CONDITIONS 40* PROBLEMS 48* REFERENCES . . . . . . . . . . . . . 51* 2 QUANTUM ERROR CORREETING CODES 57* 2.1 QUANTUM OPERATIONS . . . . . . . 57* 2.1.1 OPERATOR-SUM REPRESENTATION . 58* 2.1.2 DEPOLARIZING CHANNEL .. 70* 2.1.3 OTHER ERROR MODELS 71* 2.2 QUANTUM ERROR CORRECTING CODES: DEFINITIONS 73* 2.3 EXAMPLE: CALDERBANK-SHOR-STEANE [7, 1, 3] CODE 75* PROBLEMS 77* REFERENCES . . . . . . . . . . . . . . . 80* VII VIII QUANTUM ERROR CORRECTION AND FAULT- TOLERANT QUANTUM COMPUTING 3 QUANTUM STABILIZER CODES* 83* 3.1* GENERAL FRAMEWORK 83* 3.1.1 SUMMARY .* 83* 3.1.2 ERRORS .* 85* 3.1.3 QUANTUM ERROR CORRECTION REPRISE* 90* 3.1.4 ENCODED OPERATIONS* 91* 3.2* EXAMPLES ..... 93* 3.2.1 [5,1,3] CODE* 93* 3.2.2 [4,2,2] CODE* 95* 3.2.3 [8,3,3] CODE* 95* 3.3* ALTERNATE FORMULATION: FINITE GEOMETRY 96* 3.4* CONCATENATED CODES ..... 100* 3.4.1 SINGLE QUBIT ENCODING* 101* 3.4.2 MULTI-QUBIT ENCODING* 104* PROBLEMS 110* REFERENCES . . . . . . . . . . . 113* 4 QUANTUM STABILIZER CODES: EFLICIENT ENCODING AND DECODING115 4.1* STANDARD FORM 115* 4.2* ENCODING 121* 4.3* DECODING 127* PROBLEMS 131* REFERENCES 134* 5 FAULT-TOLERANT QUANTUM COMPUTING* 137* 5.1* FAULT-TOLERANCE . 137* 5.1.1 TO ENCODE OR NOT TO ENCODE ...* 139* 5.1.2 FAULT-TOLERANT DESIGN* 139* 5.2* ERROR CORRECTION . 141* 5.2.1 SYNDROME EXTRACTION .* 144* 5.2.2 SHOR STATE VERIFICATION* 148* 5.2.3 SYNDROME VERIFICATION* 151* 5.3* ENCODED OPERATIONS IN N(QN) N N(S) 153* 5.3.1 ACTION OF N(QN)* 155* 5.3.2 CSS CODES .* 158* 5.4* MEASUREMENT . 162* 5.5* FOUR-QUBIT INTERLUDE . . 166* 5.6* MULTI-QUBIT STABILIZER CODES 169* 5.7* OPERATIONS OUTSIDE N(QN)-TOFFOLI GATE 173* 5.8* EXAMPLE: [5,1,3] CODE . 180* 5.9* EXAMPLE: [4,2,2] CODE . 186* 5.9.1 FAULT-TOLERANT QUANTUM COMPUTING* 186* 5.9.2 FAULT-TOLERANT ENCODED OPERATIONS. 187* PROBLEMS 188* REFERENCES . . . . . . . . . . . . . . . . . . . 192* TABLE OF CONTENTS IX* 6 ACCURACY THRESHOLD THEOREM 195* 6.1 PRELIMINARIES . 195* 6.1.1 CONCATENATED QECCS 196* 6.1.2 PRINCIPAL ASSUMPTIONS 197* 6.2 THRESHOLD ANALYSIS . . . . . . 199* 6.2.1 RECURSION - RELATION FOR PW 200* 6.2.2 CLIFFORD GROUP GATES AND STORAGE REGISTERS 208* 6.2.3 TOFFOLI GATE 210* PROBLEMS 219* REFERENCES . . . . . * 220* 7 BOUNDS ON QUANTUM ERROR CORRECTING CODES 223* 7.1 QUANTUM HAMMING BOUND ..... 223* 7.2 QUANTUM GILBERT-VARSHAMOV BOUND * 225* 7.3 QUANTUM SINGLETON BOUND . * 227* 7.4 LINEAR PROGRAMMING BOUNDS FOR QECCS 229* 7.4.1 WEIGHT ENUMERATORS . * 230* 7.4.2 QUANTUM MACWILLIAMS IDENTITY . * 233* 7.4.3 SHADOW ENUMERATOR . .236* 7.4.4 BOUNDS VIA LINEAR PROGRAMMING 239* 7.5 ENTANGLEMENT PURIFICATION AND QECCS 240* 7.5.1 PURIFYING ENTANGLEMENT 241* 7.5.2 QECCS AND 1-EPPS 255* PROBLEMS 257* REFERENCES . . . . . . . . . . 258* A GROUP THEORY 261* A.1 FUNDAMENTAL NOTIONS 261* A.1.1 GROUPS ... 261* A.1.2 SUBGROUPS. 262* A.2 GROUP ACTION .. 265* A.2.1 ON A SET. 265* A.2.2 ON ITSELF . 265* A.3 MAPPING GROUPS 266* A.3.1 HOMOMORPHISMS * 266* A.3.2 FUNDAMENTAL HOMOMORPHISM THEOREM 267* REFERENCES . . . . . . . . . . . . . . . . . . . . . * 268* B QUANTUM MECHANICS 269* B.1 STATES . .270* B.2 COMPOSITE SYSTEMS 271* B.3 OBSERVABLES . * 271* B.4 DYNAMICS . * 272* B.5 MEASUREMENT AND STATE PREPARATION .273* B.6 MIXED STATES . * 276* X QUANTUM ERROR CORRECTION AND FAULT- TOLERANT QUANTUM COMPUTING REFERENCES . .* * 280* C QUANTUM CIRCUITS* 283* C.L* BASIC CIRCUIT ELEMENTS . .283* C.2* GOTTESMAN-KNILL THEOREM . * 285* C.3* UNIVERSAL SETS OF QUANTUM GATES 286* REFERENCES . . . . . . . . . . . . . * 288* 289 INDEX
adam_txt	CONTENTS LIST OF FIGURES XI* LIST OF TABLES XV* PREFACE XVII* 1 INTRODUETION 1* 1.1 HISTORICAL BACKGROUND 2* 1.2 CLASSICAL ERROR CORRECTING CODES 4* 1.2.1 LINEAR ERROR CORRECTING CODES 4* 1.2.2 ERRORS, HAMMING WEIGHT AND DISTANCE . 7* 1.2.3 ERROR DETECTION AND CORRECTION 10* 1.2.4 ERRORPROBABILITY . . . . . . . . . . . . . 12* 1.2.5 BOUNDS ON CODE PARAMETERS. . . . . . . 14* 1.3 USING QUANTUM SYSTEMS TO STORE AND PROCESS DATA 17* 1.3.1 LINEAR SUPERPOSITION AND QUANTUM PARALLELISM 17* 1.3.2 NO-CLONING THEOREM . . . . . . 20* 1.3.3 MEASUREMENT. 21* 1.3.4 DISTINGUISHING QUANTUM STATES 23* 1.3.5 ENTANGLEMENT. 24* 1.3.6 NOISE: ERRORS AND DECOHERENCE 28* 1.4 QUANTUM ERROR CORRECTING CODES-FIRST PASS 31* 1.4.1 REDUNDANCY WITHOUT CLONING . . . . 31* 1.4.2 NECESSARY AND SUFFICIENT CONDITIONS 40* PROBLEMS 48* REFERENCES . . . . . . . . . . . . . 51* 2 QUANTUM ERROR CORREETING CODES 57* 2.1 QUANTUM OPERATIONS . . . . . . . 57* 2.1.1 OPERATOR-SUM REPRESENTATION . 58* 2.1.2 DEPOLARIZING CHANNEL . 70* 2.1.3 OTHER ERROR MODELS 71* 2.2 QUANTUM ERROR CORRECTING CODES: DEFINITIONS 73* 2.3 EXAMPLE: CALDERBANK-SHOR-STEANE [7, 1, 3] CODE 75* PROBLEMS 77* REFERENCES . . . . . . . . . . . . . . . 80* VII VIII QUANTUM ERROR CORRECTION AND FAULT- TOLERANT QUANTUM COMPUTING 3 QUANTUM STABILIZER CODES* 83* 3.1* GENERAL FRAMEWORK 83* 3.1.1 SUMMARY .* 83* 3.1.2 ERRORS .* 85* 3.1.3 QUANTUM ERROR CORRECTION REPRISE* 90* 3.1.4 ENCODED OPERATIONS* 91* 3.2* EXAMPLES . 93* 3.2.1 [5,1,3] CODE* 93* 3.2.2 [4,2,2] CODE* 95* 3.2.3 [8,3,3] CODE* 95* 3.3* ALTERNATE FORMULATION: FINITE GEOMETRY 96* 3.4* CONCATENATED CODES . 100* 3.4.1 SINGLE QUBIT ENCODING* 101* 3.4.2 MULTI-QUBIT ENCODING* 104* PROBLEMS 110* REFERENCES . . . . . . . . . . . 113* 4 QUANTUM STABILIZER CODES: EFLICIENT ENCODING AND DECODING115 4.1* STANDARD FORM 115* 4.2* ENCODING 121* 4.3* DECODING 127* PROBLEMS 131* REFERENCES 134* 5 FAULT-TOLERANT QUANTUM COMPUTING* 137* 5.1* FAULT-TOLERANCE . 137* 5.1.1 TO ENCODE OR NOT TO ENCODE .* 139* 5.1.2 FAULT-TOLERANT DESIGN* 139* 5.2* ERROR CORRECTION . 141* 5.2.1 SYNDROME EXTRACTION .* 144* 5.2.2 SHOR STATE VERIFICATION* 148* 5.2.3 SYNDROME VERIFICATION* 151* 5.3* ENCODED OPERATIONS IN N(QN) N N(S) 153* 5.3.1 ACTION OF N(QN)* 155* 5.3.2 CSS CODES .* 158* 5.4* MEASUREMENT . 162* 5.5* FOUR-QUBIT INTERLUDE . . 166* 5.6* MULTI-QUBIT STABILIZER CODES 169* 5.7* OPERATIONS OUTSIDE N(QN)-TOFFOLI GATE 173* 5.8* EXAMPLE: [5,1,3] CODE . 180* 5.9* EXAMPLE: [4,2,2] CODE . 186* 5.9.1 FAULT-TOLERANT QUANTUM COMPUTING* 186* 5.9.2 FAULT-TOLERANT ENCODED OPERATIONS. 187* PROBLEMS 188* REFERENCES . . . . . . . . . . . . . . . . . . . 192* TABLE OF CONTENTS IX* 6 ACCURACY THRESHOLD THEOREM 195* 6.1 PRELIMINARIES . 195* 6.1.1 CONCATENATED QECCS 196* 6.1.2 PRINCIPAL ASSUMPTIONS 197* 6.2 THRESHOLD ANALYSIS . . . . . . 199* 6.2.1 RECURSION - RELATION FOR PW 200* 6.2.2 CLIFFORD GROUP GATES AND STORAGE REGISTERS 208* 6.2.3 TOFFOLI GATE 210* PROBLEMS 219* REFERENCES . . . . . * 220* 7 BOUNDS ON QUANTUM ERROR CORRECTING CODES 223* 7.1 QUANTUM HAMMING BOUND . 223* 7.2 QUANTUM GILBERT-VARSHAMOV BOUND * 225* 7.3 QUANTUM SINGLETON BOUND . * 227* 7.4 LINEAR PROGRAMMING BOUNDS FOR QECCS 229* 7.4.1 WEIGHT ENUMERATORS . * 230* 7.4.2 QUANTUM MACWILLIAMS IDENTITY . * 233* 7.4.3 SHADOW ENUMERATOR . .236* 7.4.4 BOUNDS VIA LINEAR PROGRAMMING 239* 7.5 ENTANGLEMENT PURIFICATION AND QECCS 240* 7.5.1 PURIFYING ENTANGLEMENT 241* 7.5.2 QECCS AND 1-EPPS 255* PROBLEMS 257* REFERENCES . . . . . . . . . . 258* A GROUP THEORY 261* A.1 FUNDAMENTAL NOTIONS 261* A.1.1 GROUPS . 261* A.1.2 SUBGROUPS. 262* A.2 GROUP ACTION . 265* A.2.1 ON A SET. 265* A.2.2 ON ITSELF . 265* A.3 MAPPING GROUPS 266* A.3.1 HOMOMORPHISMS * 266* A.3.2 FUNDAMENTAL HOMOMORPHISM THEOREM 267* REFERENCES . . . . . . . . . . . . . . . . . . . . . * 268* B QUANTUM MECHANICS 269* B.1 STATES . .270* B.2 COMPOSITE SYSTEMS 271* B.3 OBSERVABLES . * 271* B.4 DYNAMICS . * 272* B.5 MEASUREMENT AND STATE PREPARATION .273* B.6 MIXED STATES . * 276* X QUANTUM ERROR CORRECTION AND FAULT- TOLERANT QUANTUM COMPUTING REFERENCES . .* * 280* C QUANTUM CIRCUITS* 283* C.L* BASIC CIRCUIT ELEMENTS . .283* C.2* GOTTESMAN-KNILL THEOREM . * 285* C.3* UNIVERSAL SETS OF QUANTUM GATES 286* REFERENCES . . . . . . . . . . . . . * 288* 289 INDEX
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spelling	Gaitan, Frank Verfasser aut Quantum error correction and fault tolerant quantum computing Frank Gaitan Boca Raton, Fla. [u.a.] CRC Press 2008 XVIII, 292 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Quantum computers Error-correcting codes (Information theory) Fault-tolerant computing Fehlerkorrekturcode (DE-588)4124917-3 gnd rswk-swf Fehlertoleranz (DE-588)4123192-2 gnd rswk-swf Quantencomputer (DE-588)4533372-5 gnd rswk-swf Quantencomputer (DE-588)4533372-5 s Fehlertoleranz (DE-588)4123192-2 s Fehlerkorrekturcode (DE-588)4124917-3 s DE-604 OEBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016399620&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Gaitan, Frank Quantum error correction and fault tolerant quantum computing Quantum computers Error-correcting codes (Information theory) Fault-tolerant computing Fehlerkorrekturcode (DE-588)4124917-3 gnd Fehlertoleranz (DE-588)4123192-2 gnd Quantencomputer (DE-588)4533372-5 gnd
subject_GND	(DE-588)4124917-3 (DE-588)4123192-2 (DE-588)4533372-5
title	Quantum error correction and fault tolerant quantum computing
title_auth	Quantum error correction and fault tolerant quantum computing
title_exact_search	Quantum error correction and fault tolerant quantum computing
title_exact_search_txtP	Quantum error correction and fault tolerant quantum computing
title_full	Quantum error correction and fault tolerant quantum computing Frank Gaitan
title_fullStr	Quantum error correction and fault tolerant quantum computing Frank Gaitan
title_full_unstemmed	Quantum error correction and fault tolerant quantum computing Frank Gaitan
title_short	Quantum error correction and fault tolerant quantum computing
title_sort	quantum error correction and fault tolerant quantum computing
topic	Quantum computers Error-correcting codes (Information theory) Fault-tolerant computing Fehlerkorrekturcode (DE-588)4124917-3 gnd Fehlertoleranz (DE-588)4123192-2 gnd Quantencomputer (DE-588)4533372-5 gnd
topic_facet	Quantum computers Error-correcting codes (Information theory) Fault-tolerant computing Fehlerkorrekturcode Fehlertoleranz Quantencomputer
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