Mathematics of Quantum Computation

by Brylinski; Ranee K.

• ISBN13: 9781584882824
• ISBN10: 1584882824
• Edition: 1st
• Format: Hardcover
• Copyright: 2002-02-14
• Publisher: Chapman & Hall/

Summary

Among the most exciting developments in science today is the design and construction of the quantum computer. Its realization will be the result of multidisciplinary efforts, but ultimately, it is mathematics that lies at the heart of theoretical quantum computer science.Mathematics of Quantum Computation brings together leading computer scientists, mathematicians, and physicists to provide the first interdisciplinary but mathematically focused exploration of the field's foundations and state of the art. Each section of the book addresses an area of major research, and does so with introductory material that brings newcomers quickly up to speed. Chapters that are more advanced include recent developments not yet published in the open literature.Information technology will inevitably enter into the realm of quantum mechanics, and, more than all the atomic, molecular, optical, and nanotechnology advances, it is the device-independent mathematics that is the foundation of quantum computer and information science. Mathematics of Quantum Computation offers the first up-to-date coverage that has the technical depth and breadth needed by those interested in the challenges being confronted at the frontiers of research.

Table of Contents

Preface	v
Quantum Entanglement	1	(2)
Algebraic measures of entanglement	3	(22)
Jean-Luc Brylinski
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	25	(52)
Berthold-Georg Englert
Nasser Metwally
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	77	(22)
David A. Meyer
Noland Wallach
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	101	(16)
Jean-Luc Brylinski
Ranee Brylinski
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	119	(16)
Lov K. Grover
Anirvan M. Sengupta
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	135	(26)
Goong Chen
Stephen A. Fulling
Jeesen Chen
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	161	(8)
Goong Chen
Shunhua Sun
Introduction	162	(1)
Multiobject search algorithm using a general unitary transformation	163	(6)
Quantum Computational Complexity	169	(52)
Counting complexity and quantum computation	171	(50)
Stephen A. Fenner
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	223	(30)
Markus Grassl
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)
Additive quantum codes	241	(9)
Construction	241	(5)
Example: quantum Hamming code	246	(4)
Conclusions	250	(3)
Clifford codes	253	(22)
Andreas Klappenecker
Martin Rotteler
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	277	(10)
Jean-Luc Brylinski
Ranee Brylinski
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	287	(34)
Michael H. Freedman
David A. Meyer
Feng Luo
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	323	(34)
Kishore T. Kapale
M. Suhail Zubairy
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	359	(44)
Almut Beige
Berthold-Georg Englert
Christian Kurtsiefer
Harald Weinfurter
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	405	(16)
Stephen A. Fulling
Review of NMR quantum computing	406	(1)
Review of modular arithmetic	407	(2)
A proposed ``quantum'' implementation	409	(3)
Aftermath	412	(9)
Index	421