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Tables of Contents for Mathematics of Quantum Computation
Algebraic measures of entanglement
Jean-Luc Brylinski3
22
Kinematics of qubit pairs
Berthold-Georg Englert and
Nasser Metwally25
52
Basic classification of states
34
2
Projectors and subspaces
36
7
Rank 136
3
Rank 239
2
Rank 341
2
Rank 443
1
Positivity and separability
43
4
Lewenstein-Sanpera decompositions
47
13
Basic properties of optimal LSDs51
3
Optimal LSDs of truly positive states54
6
Self-transposed states60
3
Generalized Werner states63
3
States of rank 266
6
Invariants for multiple qubits: the case of 3 qubits
David A. Meyer and
Noland Wallach77
22
Invariants for compact Lie groups
79
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 Brylinski101
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
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. Sengupta119
16
Rules of the game125
1
Algorithm126
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
Counting131
1
Mechanical applications132
1
Quantum mechanical applications133
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 Chen135
26
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 Sun161
8
Multiobject search algorithm using a general unitary transformation
163
6
Quantum Computational Complexity
169
52
Counting complexity and quantum computation
Stephen A. Fenner171
50
Qubits, quantum gates, and quantum circuits175
3
Classical complexity178
14
Classical computations on a quantum circuit192
2
Relativizing quantum computation194
1
Equivalence of FQP and GapP
195
5
Strengths of the quantum model
200
7
Oracle results202
5
Limitations of the quantum model
207
7
Quantum Error-Correcting Codes
221
54
Algorithmic aspects of quantum error-correcting codes
Markus Grassl223
30
General quantum error-correcting codes
224
9
General errors224
5
Local errors229
4
Construction233
6
Example: binary Hamming code239
2
Additive quantum codes
241
9
Construction241
5
Example: quantum Hamming code246
4
Clifford codes
Andreas Klappenecker and
Martin Rotteler253
22
Quantum error control codes
256
3
Clifford codes that are stabilizer codes
265
4
A remarkable error group
269
1
Quantum Computing Algebraic and Geometric Structures
275
46
Invariant Polynomial functions on k qudits
Jean-Luc Brylinski and
Ranee Brylinski277
10
Polynomial invariants of tensor states
279
2
The generalized determinant function
281
1
Quartic invariants of k qudits
283
4
Z2-systolic freedom and quantum codes
Michael H. Freedman David A. Meyer and
Feng Luo287
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 Zubairy323
34
Teleportation of a two-state system
326
11
The formal scheme327
3
Cavity QED implementation330
7
Discrete N-state quantum systems
337
3
Entangled state teleportation
340
6
Two-qubit entangled state341
3
N-qubit entangled state344
2
Continuous quantum variable states
346
5
Nonlocal measurements346
2
Wigner functions348
3
Quantum Secure Communication and Quantum Cryptography
357
46
Communicating with qubit pairs
Almut Beige Berthold-Georg Englert Christian Kurtsiefer and
Harald Weinfurter359
44
The mean king's problem
361
10
The Vaidman-Aharonov-Albert puzzle361
1
The stranded physicist's solution362
5
The mean king's second challenge367
2
A different perspective369
2
BB84: cryptography with single qubits
371
6
Description of the scheme372
1
Eavesdropping: minimizing the error probability373
2
Eavesdropping: maximizing the raw information375
2
Cryptography with qubit pairs
377
8
Description of the scheme377
3
Eavesdropping: minimizing the error probability380
4
Eavesdropping: maximizing the raw information384
1
Idealized single-photon schemes
385
7
BB84 scheme with two state pairs385
3
Qubit-pair scheme with four state pairs388
4
Direct communication with qubit pairs
392
6
Description of the scheme392
3
Minimal error probability395
3
Commentary on Quantum Computing
403
18
Transgressing the boundaries of quantum computation: a contribution to the hermeneutics of the NMR paradigm
Stephen A. Fulling405
16
Review of NMR quantum computing
406
1
Review of modular arithmetic
407
2
A proposed ``quantum'' implementation
409
3
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