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E-raamat: Quantum Error Correction and Fault Tolerant Quantum Computing

(Laboratory for Physical Sciences, College Park, Maryland, USA)
  • Formaat: 312 pages
  • Ilmumisaeg: 03-Oct-2018
  • Kirjastus: CRC Press Inc
  • Keel: eng
  • ISBN-13: 9781351837774
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  • Formaat: 312 pages
  • Ilmumisaeg: 03-Oct-2018
  • Kirjastus: CRC Press Inc
  • Keel: eng
  • ISBN-13: 9781351837774
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It was once widely believed that quantum computation would never become a reality. However, the discovery of quantum error correction and the proof of the accuracy threshold theorem nearly ten years ago gave rise to extensive development and research aimed at creating a working, scalable quantum computer. Over a decade has passed since this monumental accomplishment yet no book-length pedagogical presentation of this important theory exists.

Quantum Error Correction and Fault Tolerant Quantum Computing offers the first full-length exposition on the realization of a theory once thought impossible. It provides in-depth coverage on the most important class of codes discovered to datequantum stabilizer codes. It brings together the central themes of quantum error correction and fault-tolerant procedures to prove the accuracy threshold theorem for a particular noise error model. The author also includes a derivation of well-known bounds on the parameters of quantum error correcting code.

Packed with over 40 real-world problems, 35 field exercises, and 17 worked-out examples, this book is the essential resource for any researcher interested in entering the quantum field as well as for those who want to understand how the unexpected realization of quantum computing is possible.
List of Figures
xi
List of Tables
xv
Preface xvii
Introduction
1(56)
Historical Background
2(2)
Classical Error Correcting Codes
4(13)
Linear Error Correcting Codes
4(3)
Errors, Hamming Weight and Distance
7(3)
Error Detection and Correction
10(2)
Error Probability
12(2)
Bounds on Code Parameters
14(3)
Using Quantum Systems to Store and Process Data
17(14)
Linear Superposition and Quantum Parallelism
17(3)
No-Cloning Theorem
20(1)
Measurement
21(2)
Distinguishing Quantum States
23(1)
Entanglement
24(4)
Noise: Errors and Decoherence
28(3)
Quantum Error Correcting Codes---First Pass
31(26)
Redundancy without Cloning
31(9)
Necessary and Sufficient Conditions
40(8)
Problems
48(3)
References
51(6)
Quantum Error Correcting Codes
57(26)
Quantum Operations
57(16)
Operator-Sum Representation
58(12)
Depolarizing Channel
70(1)
Other Error Models
71(2)
Quantum Error Correcting Codes: Definitions
73(2)
Example: Calderbank-Shor-Steane [ 7, 1, 3] Code
75(8)
Problems
77(3)
References
80(3)
Quantum Stabilizer Codes
83(32)
General Framework
83(10)
Summary
83(2)
Errors
85(5)
Quantum Error Correction Reprise
90(1)
Encoded Operations
91(2)
Examples
93(3)
[ 5,1,3] Code
93(2)
[ 4,2,2] Code
95(1)
[ 8,3,3] Code
95(1)
Alternate Formulation: Finite Geometry
96(4)
Concatenated Codes
100(15)
Single Qubit Encoding
101(3)
Multi-Qubit Encoding
104(6)
Problems
110(3)
References
113(2)
Quantum Stabilizer Codes: Efficient Encoding and Decoding
115(22)
Standard Form
115(6)
Encoding
121(6)
Decoding
127(10)
Problems
131(3)
References
134(3)
Fault-Tolerant Quantum Computing
137(58)
Fault-Tolerance
137(4)
To Encode or Not to Encode
139(1)
Fault-Tolerant Design
139(2)
Error Correction
141(12)
Syndrome Extraction
144(4)
Shor State Verification
148(3)
Syndrome Verification
151(2)
Encoded Operations in N(Gn) ∩ N (S)
153(9)
Action of N (Gn)
155(3)
CSS Codes
158(4)
Measurement
162(4)
Four-Qubit Interlude
166(3)
Multi-Qubit Stabilizer Codes
169(4)
Operations Outside N(Gn)---Toffoli Gate
173(7)
Example: [ 5,1,3] Code
180(6)
Example: [ 4,2,2] Code
186(9)
Fault-Tolerant Quantum Computing
186(1)
Fault-Tolerant Encoded Operations
187(1)
Problems
188(4)
References
192(3)
Accuracy Threshold Theorem
195(28)
Preliminaries
195(4)
Concatenated QECCs
196(1)
Principal Assumptions
197(2)
Threshold Analysis
199(24)
Recursion - Relation for pop(j)
200(8)
Clifford Group Gates and Storage Registers
208(2)
Toffoli Gate
210(9)
Problems
219(1)
References
220(3)
Bounds on Quantum Error Correcting Codes
223(38)
Quantum Hamming Bound
223(2)
Quantum Gilbert-Varshamov Bound
225(2)
Quantum Singleton Bound
227(2)
Linear Programming Bounds for QECCs
229(11)
Weight Enumerators
230(3)
Quantum MacWilliams Identity
233(3)
Shadow Enumerator
236(3)
Bounds via Linear Programming
239(1)
Entanglement Purification and QECCs
240(21)
Purifying Entanglement
241(14)
QECCs and 1-EPPs
255(2)
Problems
257(1)
References
258(3)
A. Group Theory
261(8)
Fundamental Notions
261(4)
Groups
261(1)
Subgroups
262(3)
Group Action
265(1)
On a Set
265(1)
On Itself
265(1)
Mapping Groups
266(3)
Homomorphisms
266(1)
Fundamental Homomorphism Theorem
267(1)
References
268(1)
B. Quantum Mechanics
269(14)
States
270(1)
Composite Systems
271(1)
Observables
271(1)
Dynamics
272(1)
Measurement and State Preparation
273(3)
Mixed States
276(7)
References
280(3)
C. Quantum Circuits
283(6)
Basic Circuit Elements
283(2)
Gottesman-Knill Theorem
285(1)
Universal Sets of Quantum Gates
286(3)
References
288(1)
Index 289


Gaitan, Frank