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Quantum Computing: From Linear Algebra to Physical Realizations [Hardback]

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, (Kinki University, Osaka, Japan)
  • Formāts: Hardback, 438 pages, height x width: 234x156 mm, weight: 748 g, 10 Tables, black and white; 134 Illustrations, black and white
  • Izdošanas datums: 11-Mar-2008
  • Izdevniecība: Institute of Physics Publishing
  • ISBN-10: 0750309830
  • ISBN-13: 9780750309837
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Quantum Computing: From Linear Algebra to Physical RealizationsPalielināt
  • Formāts: Hardback, 438 pages, height x width: 234x156 mm, weight: 748 g, 10 Tables, black and white; 134 Illustrations, black and white
  • Izdošanas datums: 11-Mar-2008
  • Izdevniecība: Institute of Physics Publishing
  • ISBN-10: 0750309830
  • ISBN-13: 9780750309837
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9780750309837.html
Keywords:
Covering both theory and progressive experiments, Quantum Computing: From Linear Algebra to Physical Realizations explains how and why superposition and entanglement provide the enormous computational power in quantum computing. This self-contained, classroom-tested book is divided into two sections, with the first devoted to the theoretical aspects of quantum computing and the second focused on several candidates of a working quantum computer, evaluating them according to the DiVincenzo criteria.

Topics in Part I





Linear algebra Principles of quantum mechanics Qubit and the first application of quantum information processingquantum key distribution Quantum gates Simple yet elucidating examples of quantum algorithms Quantum circuits that implement integral transforms Practical quantum algorithms, including Grovers database search algorithm and Shors factorization algorithm The disturbing issue of decoherence Important examples of quantum error-correcting codes (QECC)

Topics in Part II





DiVincenzo criteria, which are the standards a physical system must satisfy to be a candidate as a working quantum computer Liquid state NMR, one of the well-understood physical systems Ionic and atomic qubits Several types of Josephson junction qubits The quantum dots realization of qubits

Looking at the ways in which quantum computing can become reality, this book delves into enough theoretical background and experimental research to support a thorough understanding of this promising field.

Recenzijas

The book is very well structured and offers good theoretical explanations reinforced by examples. As the authors mention in the Preface, the book can be used for a quantum computing course. It is also recommended to advanced undergraduate students, postgraduate students and researchers in physics, mathematics and computer science. Zentralblatt MATH 1185

I From Linear Algebra to Quantum Computing 1
1 Basics of Vectors and Matrices
3
1.1 Vector Spaces
4
1.2 Linear Dependence and Independence of Vectors
5
1.3 Dual Vector Spaces
6
1.4 Basis, Projection Operator and Completeness Relation
8
1.4.1 Orthonormal Basis and Completeness Relation
8
1.4.2 Projection Operators
9
1.4.3 Gram-Schmidt Orthonormalization
10
1.5 Linear Operators and Matrices
11
1.5.1 Hermitian Conjugate, Hermitian and Unitary Matrices
12
1.6 Eigenvalue Problems
13
1.6.1 Eigenvalue Problems of Hermitian and Normal Matrices
14
1.7 Pauli Matrices
18
1.8 Spectral Decomposition
19
1.9 Singular Value Decomposition (SVD)
23
1.10 Tensor Product (Kronecker Product)
26
2 Framework of Quantum Mechanics
29
2.1 Fundamental Postulates
29
2.2 Some Examples
32
2.3 Multipartite System, Tensor Product and Entangled State
36
2.4 Mixed States and Density Matrices
38
2.4.1 Negativity
42
2.4.2 Partial Trace and Purification
45
2.4.3 Fidelity
47
3 Qubits and Quantum Key Distribution
51
3.1 Qubits
51
3.1.1 One Qubit
51
3.1.2 Bloch Sphere
53
3.1.3 Multi-Qubit Systems and Entangled States
54
3.1.4 Measurements
56
3.1.5 Einstein-Podolsky-Rosen (EPR) Paradox
59
3.2 Quantum Key Distribution (BB84 Protocol)
60
4 Quantum Gates, Quantum Circuit and Quantum Computation
65
4.1 Introduction
65
4.2 Quantum Gates
66
4.2.1 Simple Quantum Gates
66
4.2.2 Walsh-Hadamard Transformation
69
4.2.3 SWAP Gate and Fredkin Gate
70
4.3 Correspondence with Classical Logic Gates
71
4.3.1 NOT Gate
72
4.3.2 XOR Gate
72
4.3.3 AND Gate
73
4.3.4 OR Gate
73
4.4 No-Cloning Theorem
75
4.5 Dense Coding and Quantum Teleportation
76
4.5.1 Dense Coding
77
4.5.2 Quantum Teleportation
79
4.6 Universal Quantum Gates
82
4.7 Quantum Parallelism and Entanglement
95
5 Simple Quantum Algorithms
99
5.1 Deutsch Algorithm
99
5.2 Deutsch-Jozsa Algorithm and Bernstein-Vazirani Algorithm
101
5.3 Simon's Algorithm
105
6 Quantum Integral Transforms
109
6.1 Quantum Integral Transforms
109
6.2 Quantum Fourier Transform (QFT)
111
6.3 Application of QFT: Period-Finding
113
6.4 implementation of QFT
116
6.5 Walsh-Hadamard Transform
122
6.6 Selective Phase Rotation Transient)
123
7 Grover's Search .Algorithm
125
7.1 Searching for a Single File
125
7.2 Searching for d Files
133
8 Shor's Factorization Algorithm
137
8.1 The RSA Cryptosystem
137
8.2 Factorization Algorithm
140
8.3 Quantum Part of Shor's Algorithm
141
8.3.1 Settings for STEP 2
141
8.3.2 STEP 2
113
8.4 Probability Distribution
144
8.5 Continued Fractions and Order Finding
151
8.6 Modular Exponential Function
156
8.6.1 Adder
157
8.6.2 Modular Adder
161
8.6.3 Modular Multiplexer
166
8.6.4 Modular Exponential Function
168
8.6.5 Computational Complexity of Modular Exponential Circuit
170
9 Decoherence
173
9.1 Open Quantum System
173
9.1.1 Quantum Operations and Kraus Operators
174
9.1.2 Operator-Sum Representation and Noisy Quantum Channel
177
9.1.3 Completely Positive Maps
178
9.2 Measurements as Quantum Operations
179
9.2.1 Projective Measurements
179
9.2.2 POVM
180
9.3 Examples
181
9.3.1 Bit-Flip Channel
181
9.3.2 Phase-Flip Channel
183
9.3.3 Depolarizing Channel
185
9.3.4 Amplitude-Damping Channel
187
9.4 Lindblad Equation
188
9.4.1 Quantum Dynamical Semigroup
189
9.4.2 Lindblad Equation
189
9.4.3 Examples
192
10 Quantum Error Correcting Codes
195
10.1 Introduction
195
10.2 Three-Qubit Bit-Flip Code and Phase-Flip Code
196
10.2.1 Bit-Flip QECC
196
10.2.2 Phase-Flip QECC
202
10.3 Shor's Nine-Qubit Code
203
10.3.1 Encoding
204
10.3.2 Transmission
205
10.3.3 Error Syndrome Detection and Correction
205
10.3.4 Decoding
208
10.4 Seven-Qubit QECC
209
10.4.1 Classical Theory of Error Correcting Codes
209
10.4.2 Seven-Qubit QECC
213
10.4.3 Gate Operations for Seven-Qubit QECC
220
10.5 Five-Qubit QECC
224
10.5.1 Encoding
224
10.5.2 Error Syndrome Detection
227
II Physical Realizations of Quantum Computing 231
11 DiVincenzo Criteria
233
11.1 Introduction
233
11.2 DiVincenzo Criteria
234
11.3 Physical Realizations
239
12 NMR Quantum Computer
241
12.1 Introduction
241
12.2 NMR Spectrometer
241
12.2.1 Molecules
242
12.2.2 NMR Spectrometer
242
12.3 Hamiltonian
245
12.3.1 Single-Spin Hamiltonian
245
12.3.2 Multi-Spin Hamiltonian
248
12.4 Implementation of Gates and Algorithms
252
12.4.1 One-Qubit Gates in One-Qubit Molecule
252
12.4.2 One-Qubit Operation in Two-Qubit Molecule: Bloch-Siegert Effect
256
12.4.3 Two-Qubit Gates
257
12.4.4 Multi-Qubit Gates
259
12.5 Time-Optimal Control of NMR Quantum Computer
262
12.5.1 A Brief Introduction to Lie Algebras and Lie Groups
262
12.5.2 Cartan Decomposition and Optimal Implementation of Two-Qubit Gates
264
12.6 Measurements
268
12.6.1 Introduction and Preliminary
268
12.6.2 One-Qubit Quantum State Tomography
269
12.6.3 Free Induction Decay (FID)
270
12.6.4 Two-Qubit Tomography
271
12.7 Preparation of Pseudopure State
274
12.7.1 Temporal Averaging
276
12.7.2 Spatial Averaging
277
12.8 DiVincenzo Criteria
281
13 Trapped Ions
285
13.1 Introduction
285
13.2 Electronic States of Ions as Quoits
287
13.3 Ions in Paul Trap
289
13.3.1 Trapping Potential
289
13.3.2 Lattice Formation
294
13.3.3 Normal Modes
296
13.4 Ion Qubit
298
13.4.1 One-Spin Hamiltonian
298
13.4.2 Sideband Cooling
301
13.5 Quantum Gates
302
13.5.1 One-Qubit Gates
302
13.5.2 CNOT Gate
304
13.6 Readout
306
13.7 DiVincenzo Criteria
307
14 Quantum Computing with Neutral Atoms
311
14.1 Introduction
311
14.2 Trapping Neutral Atoms
311
14.2.1 Alkali Metal Atoms
311
14.2.2 Magneto-Optical Trap (MOT)
312
14.2.3 Optical Dipole Trap
314
14.2.4 Optical Lattice
316
14.2.5 Spin-Dependent Optical Potential
317
14.3 One-Qubit Gates
319
14.4 Quantum State Engineering of Neutral Atoms
321
14.4.1 Trapping of a Single Atom
321
14.4.2 Rabi Oscillation
321
14.4.3 Neutral Atom Quantum Regisiter
323
14.5 Preparation of Entangled Neutral Atoms
324
14.6 DiVincenzo Criteria
327
15 Josephson Junction Qubits
329
15.1 Introduction
329
15.2 Nanoscale Josephson Junctions and SQUIDs
330
15.2.1 Josephson Junctions
330
15.2.2 SQUIDs
333
15.3 Charge Qubit
337
15.3.1 Simple Cooper Pair Box
337
15.3.2 Split Cooper Pair Box
341
15.4 Flux Qubit
342
15.4.1 Simplest Flux Qubit
342
15.4.2 Three-Junction Flux Qubit
345
15.5 Quantronium
347
15.6 Current-Biased Qubit (Phase Qubit)
348
15.7 Readout
352
15.7.1 Charge Qubit
352
15.7.2 Readout of Quantronium
355
15.7.3 Switching Current Readout of Flux Qubits
357
15.8 Coupled Qubits
358
15.8.1 Capacitively Coupled Charge Qubits
359
15.8.2 Inductive Coupling of Charge Qubits
362
15.8.3 Tunable Coupling between Flux Qubits
366
15.8.4 Coupling Flux Qubits with an LC Resonator
369
15.9 DiVincenzo Criteria
374
16 Quantum Computing with Quantum Dots
377
16.1 Introduction
377
16.2 Mesoscopic Semiconductors
377
16.2.1 Two-Dimensional Electron Gas in Inversion Layer
377
16.2.2 Coulomb Blockade
378
16.3 Electron Charge Qubit
383
16.3.1 Electron Charge Qubit
384
16.3.2 Rabi Oscillation
385
16.4 Electron Spin Qubit
386
16.4.1 Electron Spin Qubit
386
16.4.2 Single-Qubit Operations
387
16.4.3 Coherence Time
390
16.5 DiVincenzo Criteria
396
16.5.1 Charge Qubits
396
16.5.2 Spin Qubits
397
A Solutions to Selected Exercises 399
Index 417
Mikio Nakahara, Tetsuo Ohmi