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Tables of Contents for Mathematics of Quantum Computation
Chapter/Section Title
Page #
Page Count
Preface
v
Quantum Entanglement
1
2
Algebraic measures of entanglement
Jean-Luc Brylinski
3
22
Introduction
3
1
Rank of a tensor
4
4
Tensors in (C2)2
8
1
Tensor in (C2)3
8
10
Tensors in (C2)4
18
7
Kinematics of qubit pairs
Berthold-Georg Englert and Nasser Metwally
25
52
Introduction
25
3
Preliminaries
28
6
Basic classification of states
34
2
Projectors and subspaces
36
7
Rank 1
36
3
Rank 2
39
2
Rank 3
41
2
Rank 4
43
1
Positivity and separability
43
4
Lewenstein-Sanpera decompositions
47
13
Basic properties of optimal LSDs
51
3
Optimal LSDs of truly positive states
54
6
Examples
60
12
Self-transposed states
60
3
Generalized Werner states
63
3
States of rank 2
66
6
Acknowledgments
72
5
Invariants for multiple qubits: the case of 3 qubits
David A. Meyer and Noland Wallach
77
22
Introduction
77
2
Invariants for compact Lie groups
79
3
The simplest cases
82
3
The case of 3 qubits
85
3
A basic set of invariants for 3 qubits
88
7
Some implications for other representations
95
4
Universality of Quantum Gates
99
18
Universal quantum gates
Jean-Luc Brylinski and Ranee Brylinski
101
16
Statements of main results
101
3
Examples and relations to works of other authors
104
2
Proof of theorem 4.1 (outline)
106
1
First step: from universality to exact universality
107
1
Second step: reduction to n = 2
108
1
Fourth Step: analyzing the Lie algebra g
109
1
Fifth Step: the normalizer of H
110
2
Proof of theorem 4.2
112
1
A variant of theorem 4.1
113
4
Quantum Search Algorithms
117
52
From coupled pendulums to quantum search
Lov K. Grover and Anirvan M. Sengupta
119
16
Introduction
120
1
Classical analogy
120
1
N Coupled pendulums
121
4
The algorithm
125
2
Rules of the game
125
1
Algorithm
126
1
Towards quantum searching
127
1
The quantum search algorithm
128
2
Why does it take O(√N) cycles?
130
1
Applications and extensions
131
4
Counting
131
1
Mechanical applications
132
1
Quantum mechanical applications
133
2
Generalization of Grover's algorithm to multiobject search in quantum computing, Part I: continuous time and discrete time
Goong Chen Stephen A. Fulling and Jeesen Chen
135
26
Introduction
135
2
Continuous time quantum computing algorithm for multiobject search
137
10
Discrete time case: straightforward generalization of Grover's algorithm to multiobject search
147
14
Generalization of Grover's algorithm to multiobject search in quantum computing, Part II: general unitary transformations
Goong Chen and Shunhua Sun
161
8
Introduction
162
1
Multiobject search algorithm using a general unitary transformation
163
6
Quantum Computational Complexity
169
52
Counting complexity and quantum computation
Stephen A. Fenner
171
50
Introduction
171
2
Preliminaries
173
22
Qubits, quantum gates, and quantum circuits
175
3
Classical complexity
178
14
Classical computations on a quantum circuit
192
2
Relativizing quantum computation
194
1
Equivalence of FQP and GapP
195
5
Strengths of the quantum model
200
7
Oracle results
202
5
Limitations of the quantum model
207
7
Conclusions
214
7
Quantum Error-Correcting Codes
221
54
Algorithmic aspects of quantum error-correcting codes
Markus Grassl
223
30
Introduction
223
1
General quantum error-correcting codes
224
9
General errors
224
5
Local errors
229
4
Binary quantum codes
233
8
Construction
233
6
Example: binary Hamming code
239
2
241
9
Construction
241
5
Example: quantum Hamming code
246
4
Conclusions
250
3
Clifford codes
Andreas Klappenecker and Martin Rotteler
253
22
Introduction
253
2
Motivation
255
1
Quantum error control codes
256
3
Nice error bases
259
3
Stabilizer codes
262
2
Clifford codes
264
1
Clifford codes that are stabilizer codes
265
4
A remarkable error group
269
1
A weird error group
269
1
Conclusions
270
5
Quantum Computing Algebraic and Geometric Structures
275
46
Invariant Polynomial functions on k qudits
Jean-Luc Brylinski and Ranee Brylinski
277
10
Introduction
277
2
Polynomial invariants of tensor states
279
2
The generalized determinant function
281
1
Asymptotics as k → ∞
282
1
Quartic invariants of k qudits
283
4
Z2-systolic freedom and quantum codes
Michael H. Freedman David A. Meyer and Feng Luo
287
34
Preliminaries and statement of results
287
7
Mapping torus constructions
294
7
Verification of freedom and curvature estimates
301
7
Quantum codes from Riemannian manifolds
308
13
Quantum Teleportation
321
36
Quantum teleportation
Kishore T. Kapale and M. Suhail Zubairy
323
34
Introduction
323
3
Teleportation of a two-state system
326
11
The formal scheme
327
3
Cavity QED implementation
330
7
Discrete N-state quantum systems
337
3
Entangled state teleportation
340
6
Two-qubit entangled state
341
3
N-qubit entangled state
344
2
Continuous quantum variable states
346
5
Nonlocal measurements
346
2
Wigner functions
348
3
Concluding remarks
351
6
Quantum Secure Communication and Quantum Cryptography
357
46
Communicating with qubit pairs
Almut Beige Berthold-Georg Englert Christian Kurtsiefer and Harald Weinfurter
359
44
Introduction
360
1
The mean king's problem
361
10
The Vaidman-Aharonov-Albert puzzle
361
1
The stranded physicist's solution
362
5
The mean king's second challenge
367
2
A different perspective
369
2
BB84: cryptography with single qubits
371
6
Description of the scheme
372
1
Eavesdropping: minimizing the error probability
373
2
Eavesdropping: maximizing the raw information
375
2
Cryptography with qubit pairs
377
8
Description of the scheme
377
3
Eavesdropping: minimizing the error probability
380
4
Eavesdropping: maximizing the raw information
384
1
Idealized single-photon schemes
385
7
BB84 scheme with two state pairs
385
3
Qubit-pair scheme with four state pairs
388
4
Direct communication with qubit pairs
392
6
Description of the scheme
392
3
Minimal error probability
395
3
Acknowledgments
398
5
Commentary on Quantum Computing
403
18
Transgressing the boundaries of quantum computation: a contribution to the hermeneutics of the NMR paradigm
Stephen A. Fulling
405
16
Review of NMR quantum computing
406
1
Review of modular arithmetic
407
2
A proposed ``quantum'' implementation
409
3
Aftermath
412
9
Index
421
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