Mathematical Aspects of Quantum Computing 2007.

Nakahara, Mikio.

Mathematical Aspects of Quantum Computing 2007. - 1 online resource (240 pages) - Kinki University Series on Quantum Computing Ser. ; v.1 . - Kinki University Series on Quantum Computing Ser. .

Intro -- CONTENTS -- Preface -- LIST OF PARTICIPANTS -- Quantum Computing: An Overview M. Nakahara -- 1. Introduction -- 2. Quantum Physics -- 2.1. Notation and conventions -- 2.2. Axioms of quantum mechanics -- 2.3. Simple example -- 2.4. Multipartite system, tensor product and entangled state -- 2.5. Mixed states and density matrices -- 2.6. Negativity -- 2.7. Partial trace and purification -- 3. Qubits -- 3.1. One qubit -- 3.2. Bloch sphere -- 3.3. Multi-qubit systems and entangled states -- 4. Quantum Gates, Quantum Circuit and Quantum Computation -- 4.1. Introduction -- 4.2. Quantum gates -- 4.2.1. Simple quantum gates -- 4.2.2. Walsh-Hadamard transformation -- 4.2.3. SWAP gate and Fredkin gate -- 4.3. No-cloning theorem -- 4.4. Quantum teleportation -- 4.5. Universal quantum gates -- 4.6. Quantum parallelism and entanglement -- 5. Simple Quantum Algorithms -- 5.1. Deutsch algorithm -- 5.2. Deutsch-Jozsa algorithm -- 6. Decoherence -- 6.1. Open quantum system -- 6.1.1. Quantum operations and Kraus operators -- 6.1.2. Operator-sum representation and noisy quantum channel -- 6.1.3. Completely positive maps -- 6.2. Measurements as quantum operations -- 6.2.1. Projective measurements -- 6.2.2. POVM -- 6.3. Examples -- 6.3.1. Bit- flip channel -- 6.3.2. Phase-flip channel -- 7. Quantum Error Correcting Codes -- 7.1. Introduction -- 7.2. Three-qubit bit-flip code: the simplest example -- 7.2.1. Bit-flip QECC -- 7.2.2. Encoding -- 7.2.3. Transmission -- 7.2.4. Error syndrome dectection and correction -- 7.2.5. Decoding -- 7.2.6. Miracle of entanglement -- 7.2.7. Continuous rotations -- 8. DiVincenzo Criteria -- 8.1. DiVincenzo criteria -- 8.2. Physical realizations -- Acknowledgements -- References -- Braid Group and Topological Quantum Computing T. Ootsuka, K. Sakuma -- 1. Introduction -- 2. Braid Groups -- 3. Knots De.ned by Braids. 4. Topological Quantum Computing -- 5. Anyon Model -- 6. Fibonacci Anyons -- Appendix A. Fundamental group -- References -- An Introduction to Entanglement Theory D. J. H. Markham -- 1. Introduction -- 2. Quantum Mechanics and State Space -- 2.1. State space -- 2.2. Evolution -- 2.3. POVMs, projective measurement and observables -- 2.4. Composite systems -- 3. Entanglement and Separability -- 4. Quanti.cation of Entanglement -- 4.1. Local operations and classical communication -- 4.2. Entanglement measures -- 4.3. Uniqueness of measures, order on states -- 4.4. Measuring entanglement -- 4.5. Multipartite entanglement -- 5. Conclusions -- References -- Holonomic Quantum Computing and Its Optimization S. Tanimura -- 1. Introduction -- 2. Holonomies in Mathematics and Physics -- 2.1. Holonomy in Riemannian geometry -- 2.2. Berry phase in quantum mechanics -- 2.3. Wilczek-Zee holonomy in quantum mechanics -- 2.4. Examples -- 2.4.1. Berry phase -- 2.4.2. Λ-type system -- 3. Holonomic Quantum Computer -- 4. Formulation of the Problem and its Solution -- 4.1. Geometrical setting -- 4.2. The isoholonomic problem -- 4.3. The solution: horizontal extremal curve -- 5. The Boundary-Value Problem -- 5.1. Equivalence class -- 5.2. U(1) holonomy -- 5.3. U(k) holonomy -- 6. Examples of Unitary Gates -- 6.1. Hadamard gate -- 6.2. DFT2 gate -- 7. Discussions -- 7.1. Restricted control parameters -- 7.2. Implementation -- Acknowledgements -- References -- Playing Games in Quantum Mechanical Settings: Features of Quantum Games S¸. K. ¨O zdemir, J. Shimamura, N. Imoto -- 1. Introduction -- 2. A Brief Review of Classical Game Theory -- 3. Basic Concepts of Quantum Mechanics for a Study of Games -- 3.1. Quantum bits -- 3.2. Density matrices -- 3.3. Unitary transformation -- 3.4. Measurement -- 3.5. Correlations in quantum mechanical systems. 4. A Brief Review of Quantum Game Theory: Models and Present Status -- 4.1. Coin flip game -- 4.2. Eisert's model - Prisoners' Dilemma -- 4.3. Present status in quantum game theory -- 5. Study of a Discoordination Game - Samaritan's Dilemma - Using Eisert's Model: E.ects of Shared Correlation on the Game Dynamics -- 5.1. Quantum operations and quantum correlations -- 5.2. Quantum operations and classical correlations -- 5.3. Classical operations and classical correlations -- 5.4. Classical operations and quantum correlations -- 5.5. Classical Bob versus quantum Alice -- 6. Decoherence and Quantum Version of Classical Games -- 7. Entanglement and Reproducibility of Multiparty Classical Games in Quantum Mechanical Settings -- 7.1. Reproducibility criterion to play games in quantum mechanical settings -- 7.2. Entangled states and strong reproducibility criterion -- 8. Conclusion -- Acknowledgements -- References -- Quantum Error-Correcting Codes M. Hagiwara -- 1. Introduction -- 2. Classical Error-Correcting Codes -- 2.1. Introduction to classical error-correcting codes -- 2.2. Linear codes and parity-check matrices -- 2.3. Minimum distance and [n, k, d] codes -- 2.4. Encoding -- 2.5. Decoding -- 2.6. Bit error-rate and block error-rate -- 3. How to Generalize Classical Error-Correcting Codes to Quantum Codes -- 3.1. Parity-check measurements and stabilizer codes -- 3.2. Encoding -- 3.3. Decoding -- 4. CSS Codes -- 4.1. Twisted condition -- 4.2. Characterization of code space -- 4.3. Correctable error -- 4.4. 7-Qubit code (quantum Hamming code) -- 5. Quantum LDPC Codes -- 5.1. LDPC codes and sum-product decoding -- 5.2. Application of the sum-product algorithm for the error-correction of CSS codes -- 5.3. Classical and quantum quasi-cyclic LDPC codes -- 5.4. Twisted condition for quasi-cyclic LDPC codes. 5.5. CSS QC-LDPC codes from right-shifted matrices -- 5.6. MacKay's code -- 5.7. Regular LDPC codes -- 5.8. Construction of CSS LDPC codes -- 5.9. Performance of error-correction and minimum distance -- 6. Postscript -- Acknowledgments -- References -- -Poster Summaries- -- Controled Teleportation of an Arbitrary Unknown Two-Qubit Entangled State V. Ebrahimi, R. Rahimi, M. Nakahara -- 1. Introduction -- 2. Controlled Teleportation of Bipartite Entanglement -- References -- Notes on the D¨ur-Cirac Classi.cation Y. Ota, M. Yoshida, I. Ohba -- 1. Introduction -- 2. Results -- 3. Summary -- References -- Bang-Bang Control of Entanglement in Spin-Bus-Boson Model R. Rahimi, A. SaiToh, M. Nakahara -- 1. Introduction -- 2. The Model of Decoherence -- 3. Results -- References -- Numerical Computation of Time-Dependent Multipartite Nonclassical Correlation A. SaiToh, R. Rahimi, M. Nakahara, M. Kitagawa -- References -- On Classical No-Cloning Theorem Under Liouville Dynamics and Distances T. Yamano, O. Iguchi -- Acknowledgments -- References.

This book provides a comprehensive overview of the mathematical aspects of quantum computing. It will be useful for graduate students and researchers interested in quantum computing from different areas of physics, mathematics, informatics and computer science. The lecture notes in this volume are written in a self-contained style, and hence are accessible for graduate students and researchers with even less background in the topics. Sample Chapter(s). Quantum Computing: An Overview (804 KB). Contents: Quantum Computing: An Overview (M Nakahara); Braid Group and Topological Quantum Computing (T Ootsuka & K Sakuma); An Introduction to Entanglement Theory (D J H Markham); Holonomic Quantum Computing and Its Optimization (S Tanimura et al.); Playing Games in Quantum Mechanical Settings: Features of Quantum Games (Ö K Özdemir et al.); Quantum Error-Correcting Codes (M Hagiwara); Controled Teleportation of an Arbitrary Unknown Two-Qubit Entangled State (V Ebrahimi et al.); Notes on the Dür-Cirac Classification (Y Ota et al.); Bang-Bang Control of Entanglement in Spin-Bus-Boson Model (R Rahimi et al.); Numerical Computation of Time-Dependent Multipartite Nonclassical Correlation (A SaiToh et al.); On Classical No-Cloning Theorem Under Liouville Dynamics and Distances (T Yamano & O Iguchi). Readership: Advanced undergraduate students, graduate students and researchers in physics, mathematics, informatics and computer science.

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Electronic books.

QA76.889.M37 2008

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