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E-raamat: Quantum Computation with Topological Codes: From Qubit to Topological Fault-Tolerance

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Quantum Computation with Topological Codes: From Qubit to Topological Fault-ToleranceSuurem pilt
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This book presents a self-consistent review of quantum computation with topological quantum codes. The book covers everything required to understand topological fault-tolerant quantum computation, ranging from the definition of the surface code to topological quantum error correction and topological fault-tolerant operations. The underlying basic concepts and powerful tools, such as universal quantum computation, quantum algorithms, stabilizer formalism, and measurement-based quantum computation, are also introduced in a self-consistent way. The interdisciplinary fields between quantum information and other fields of physics such as condensed matter physics and statistical physics are also explored in terms of the topological quantum codes. This book thus provides the first comprehensive description of the whole picture of topological quantum codes and quantum computation with them.

Arvustused

The work provides a good reference for quantum computation and quantum information courses, allowing for students to become familiar with major points on the quantum information theoretical aspects of topological quantum computation and the advantages of topological quantum computation for quantum noise resistance. The book is also of interest to anyone doing research on quantum computation, quantum information and quantum error correction. (Carlos Pedro Gonçalves, zbMATH 1339.81005, 2016)

1 Introduction to Quantum Computation
1(23)
1.1 Quantum Bit and Elementary Operations
1(3)
1.2 The Solovay--Kitaev Algorithm
4(2)
1.3 Multi-Qubit Gates
6(4)
1.4 Universal Quantum Computation
10(2)
1.5 Quantum Algorithms
12(9)
1.5.1 Indirect Measurement and the Hadamard Test
12(2)
1.5.2 Phase Estimation, Quantum Fourier Transformation, and Factorization
14(2)
1.5.3 A Quantum Algorithm to Approximate Jones Polynomial
16(5)
1.6 Quantum Noise
21(2)
1.7 Summary and Discussion
23(1)
2 Stabilizer Formalism and Its Applications
24(32)
2.1 Stabilizer Formalism
24(2)
2.2 Clifford Operations
26(3)
2.3 Pauli Basis Measurements
29(1)
2.4 Gottesman--Knill Theorem
30(2)
2.5 Graph States
32(3)
2.6 Measurement-Based Quantum Computation
35(8)
2.7 Quantum Error Correction Codes
43(6)
2.8 Magic State Distillation
49(6)
2.8.1 Knill-Laflamme-Zurek Protocol
49(2)
2.8.2 Bravyi-Kitaev Protocol
51(4)
2.9 Summary and Discussion
55(1)
3 Topological Stabilizer Codes
56(30)
3.1 Z2 Chain Complex
57(3)
3.2 A Bit-Flip Code: Exercise
60(1)
3.3 Definition of Surface Codes
61(4)
3.3.1 Surface Code on a Torus: Toric Code
61(3)
3.3.2 Planar Surface Code
64(1)
3.4 Topological Quantum Error Correction
65(6)
3.5 Error Correction and Spin Glass Model
71(5)
3.6 Other Topological Codes
76(3)
3.7 Connection to Topological Order in Condensed Matter Physics
79(5)
3.8 Summary and Discussion
84(2)
4 Topological Quantum Computation with Surface Codes
86(21)
4.1 Defect Pair Qubits
86(2)
4.2 Defect Creation, Annihilation, and Movement
88(3)
4.3 Logical Controlled-NOT Gate by Braiding
91(2)
4.4 Magic State Injections and Distillation
93(3)
4.5 Topological Calculus
96(6)
4.6 Faulty Syndrome Measurements and Noise Thresholds
102(4)
4.7 Summary and Discussion
106(1)
5 Topologically Protected Measurement-Based Quantum Computation
107(15)
5.1 Topological Cluster State in Three Dimensions
107(2)
5.2 Vacuum, Defect, and Singular Qubit Regions
109(1)
5.3 Elementary Operations in Topological Measurement-Based Quantum Computation
110(5)
5.4 Topological Quantum Error Correction in Three Dimensions
115(1)
5.5 Applications for Measurement-Based Quantum Computation on Thermal States
116(5)
5.6 Summary and Discussion
121(1)
Appendix A Fault-Tolerant Quantum Computation 122(5)
Appendix B Decoding Stabilizer Codes 127(4)
References 131(6)
Index 137
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