Quantum Error Correction and Fault Tolerant Quantum Computing
Frank Gaitan
Southern Illinois University
DescriptionIt was once widely believed that quantum computation would never become a reality. However, the discovery of quantum error correction and the proof of the accuracy threshold theorem nearly ten years ago gave rise to extensive development and research aimed at creating a working, scalable quantum computer. Over a decade has passed since this monumental accomplishment yet no book-length pedagogical presentation of this important theory exists. Quantum Error Correction and Fault Tolerant Quantum Computing offers the first full-length exposition on the realization of a theory once thought impossible. It provides in-depth coverage on the most important class of codes discovered to date-quantum stabilizer codes. It brings together the central themes of quantum error correction and fault-tolerant procedures to prove the accuracy threshold theorem for a particular noise error model. The author also includes a derivation of well-known bounds on the parameters of quantum error correcting code. Packed with over 40 real-world problems, 35 field exercises, and 17 worked-out examples, this book is the essential resource for any researcher interested in entering the quantum field as well as for those who want to understand how the unexpected realization of quantum computing is possible. Table of ContentsIntroductionHistorical Background Classical Error Correcting Codes Using Quantum Systems to Store and Process Data Quantum Error Correcting Codes-First Pass Quantum Error Correcting Codes Quantum Operations Quantum Error Correcting Codes: Definitions Example: Calderbank-Shor-Steane [7, 1, 3] Code Quantum Stabilizer Codes General Framework Examples Alternate Formulation: Finite Geometry Concatenated Codes Quantum Stabilizer Codes: Efficient Encoding and Decoding Standard Form Encoding Decoding Fault-Tolerant Quantum Computing Fault-Tolerance Error Correction Encoded Operations in N(Qn) n N(S) Measurement Four-Qubit Interlude Multi-Qubit Stabilizer Codes Operations Outside N(Qn)-Toffoli Gate Example: [5, 1, 3J] Code Example: [4, 2, 2J] Code Accuracy Threshold Theorem Preliminaries Threshold Analysis Bounds on Quantum Error Correcting Codes Quantum Hamming Bound Quantum Gilbert-Varshamov Bound Quantum Singleton Bound Linear Programming Bounds for QECCs Entanglement Purification and QECCs Appendix A: Group Theory Fundamental Notions Group Action Mapping Groups Appendix B: Quantum Mechanics States Composite Systems Observables Dynamics Measurement and State Preparation Mixed States References Appendix Contributors |
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