Preface -- PART I: QUANTUM ENTANGLEMENT -- ALGEBRAIC MEASURES OF ENTANGLEMENT, Jean-Luc Brylinski -- Introduction -- Rank of a Tensor -- Tensors in (C 2)A2 -- Tensors in (C 2)A3 -- Tensors in (C 2)A4 -- KINEMATICS OF QUBIT PAIRS, Berthold-Geor Englert and Nasser Metwally -- Introduction -- Basic Classification of States -- Projectors and Subspaces -- Positivity and Separability -- Lewenstein-Sanpera Decompositions -- Examples -- INVARIANTS FOR MULTIPLE QUBITS: The Case of 3 Qubits, David A. Meyer and Noland Wallach -- Introduction -- Invariants for Compact Lie Groups -- The Simplest Cases -- The Case of Three Qubits -- A Basic Set of Invariants for Three Qubits -- Some Implications for Other Representations -- PART II: UNIVERSALITY OF QUANTUM GATES -- UNIVERSAL QUANTUM GATES, Jean-Luc Brylinski and Ranee Brylinski -- Statements of Main Results -- Examples and Relations to Works of Other Authors -- From Universality to Exact Universality -- Analyzing the Lie Algebra g -- Normalizer of H -- PART III: QUANTUM SEARCH ALGORITHMS -- FROM COUPLED PENDULUMS TO QUANTUM SEARCH Lov K. Grover and Anirvan M. Sengupta -- Introduction -- Classical Analogy -- N Coupled Pendulums -- The Algorithm -- Towards Quantum Searching -- The Quantum Search Algorithm -- Why Does it Take O(vN) cycles? -- Applications and Extensions -- GENERALIZATION OF GROVER'S ALGORITHM TO MULTIOBJECT SEARCH IN QUANTUM COMPUTING, Part I: Continuous Time and Discrete Time, Goon Chen, Stephen A,. Fulling, and Jeesen Chen -- Introduction -- Analog Multiobject Quantum Search Algorithm -- Discrete Time or "Digital " Case -- GENERALIZATION OF GROVER'S ALGORITHM TO MULTIOBJECT SEARCH IN QUANTUM COMPUTING, Part II: General Unitary Transformations, Goon Chen and Shunhua Sun -- Introduction -- Multiobject Search Algorithm -- PART III: QUANTUM COMPUTATIONAL COMPLEXITY -- COUNTING COMPLEXITY AND QUANTUM COMPUTATION, Stephen A. Fenner -- Introduction -- Equivalence of FQP and GapP -- Strengths of the Quantum Model -- Limitations of the Quantum Model -- PART IV: QUANTUM ERROR-CORRECTING CODES -- ALGORITHMIC ASPECTS OF QUANTUM ERROR-CORRECTING CODES, Markus Grassl -- Introduction -- General Quantum Error-Correcting Codes -- Binary Quantum Codes -- Additive Quantum Codes -- Conclusions -- CLIFFORD CODES, Andreas Klappenecker and Martin Rotteler -- Motivation -- Quantum Error Control Codes -- Nice Error Bases -- Stabilizer Codes -- Clifford Codes -- Clifford Codes that are Stabilizer Codes -- A Remarkable Error Group -- A Weird Error Group -- Conclusions -- PART V: QUANTUM COMPUTING ALGEBRAIC AND GEOMETRIC STRUCTURES -- INVARIANT POLYNOMIAL FUNCTIONS ON K QUDITS, Jean-Luc Brylinski and Ranee Brylinski -- Introduction -- Polynomial Invariants of Tensor States -- The Generalized Determinant Function -- Asymptotics as k ®8 -- Quartic Invariants of k Qubits -- Zs-SYSTOLIC FREEDOM AND QUANTUM CODES, Michael H. Freedman, David A. Meyer, and Feng Luo -- Preliminaries and Statement of Results -- Mapping Torus Constructions -- Verification of Freedom and Curvature Estimates -- Quantum Codes from Riemannian Manifolds -- PART VI: QUANTUM TELEPORTATION, Kishore T. Kapale and M. Suhail Zubairy -- Introduction -- Teleportation of a 2-State System -- Discrete N-State Quantum Systems -- Quantum Teleportation of Entangled State -- Continuous Quantum Variable States -- Concluding Remarks -- PART VII: QUANTUM SECURE COMMUNICATION AND QUANTUM CRYPTOGRAPHY -- COMMUNICATING WITH QUBIT PAIRS, Almut Beige, Berthold-Georg Engler, Christian Kurtsiefer, and Harald Weinfurter -- Introduction -- The Mean King's Problem -- Cryptography with Single Qubits -- Cryptography with Qubit Pairs -- Idealized Single-Photon Schemes -- Direct Communication with Qubit Pairs -- PART VIII: COMMENTARY ON QUANTUM COMPUTING -- TRANSGRESSING THE BOUNDARIES OF QUANTUM COMPUTATION: A CONTRIBUTION TO THE HERMENEUTICS OF THE NMR PARADIGM, Stephen A. Fulling -- Review of NMR Quantum Computing -- Review of Modular Arithmetic -- A Proposed "Quantum " Implementation -- Aftermath -- Keywords: Nanoscience, Nanotechnology.