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arXiv:quant-ph/0007045

A Rosetta Stone for Quantum Mechanics with an Introduction to Quantum Computation

Samuel J. Lomonaco, Jr. .

Published Jul 17, 2000• 1 citation

Abstract

The purpose of these lecture notes is to provide readers, who have some mathematical background but little or no exposure to quantum mechanics and quantum computation, with enough material to begin reading the research literature in quantum computation and quantum information theory. This paper is a written version of the first of eight one hour lectures given in the American Mathematical Society (AMS) Short Course on Quantum Computation held in conjunction with the Annual Meeting of the AMS in Washington, DC, USA in January 2000, and will appear in the AMS PSAPM volume entitled "Quantum Computation." Part 1 of the paper is an introduction the to the concept of the qubit. Part 2 gives an introduction to quantum mechanics covering such topics as Dirac notation, quantum measurement, Heisenberg uncertainty, Schrodinger's equation, density operators, partial trace, multipartite quantum systems, the Heisenberg versus the Schrodinger picture, quantum entanglement, EPR paradox, quantum entropy. Part 3 gives a brief introduction to quantum computation, covering such topics as elementary quantum computing devices, wiring diagrams, the no-cloning theorem, quantum teleportation, Shor's algorithm, Grover's algorithm. Many examples are given. A table of contents as well as an index are provided for readers who wish to "pick and choose." Since this paper is intended for a diverse audience, it is written in an informal style at varying levels of difficulty and sophistication from the very elementary to the more advanced.

DOI: 10.48550/arxiv.quant-ph/0007045

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The purpose of these lecture notes is to provide readers, who have some mathematical background but little or no exposure to quantum mechanics and quantum computation, with enough material to begin reading the research literature in quantum computation and quantum information theory. This paper is a written version of the first of eight one hour lectures given in the American Mathematical Society (AMS) Short Course on Quantum Computation held in conjunction with the Annual Meeting of the AMS in Washington, DC, USA in January 2000, and will appear in the AMS PSAPM volume entitled "Quantum Computation." Part 1 of the paper is an introduction the to the concept of the qubit. Part 2 gives an introduction to quantum mechanics covering such topics as Dirac notation, quantum measurement, Heisenberg uncertainty, Schrodinger's equation, density operators, partial trace, multipartite quantum systems, the Heisenberg versus the Schrodinger picture, quantum entanglement, EPR paradox, quantum entropy. Part 3 gives a brief introduction to quantum computation, covering such topics as elementary quantum computing devices, wiring diagrams, the no-cloning theorem, quantum teleportation, Shor's algorithm, Grover's algorithm. Many examples are given. A table of contents as well as an index are provided for readers who wish to "pick and choose." Since this paper is intended for a diverse audience, it is written in an informal style at varying levels of difficulty and sophistication from the very elementary to the more advanced.

An invitation to quantum computation: A Rosetta stone for quantum mechanics with an introduction to quantum computation by S. J. Lomonaco, Jr. Qubit devices by H. E. Brandt Quantum algorithms and quantum complexity theory: Introduction to quantum algorithms by P. W. Shor Shor's quantum factoring algorithm by S. J. Lomonaco, Jr. Grover's quantum search algorithm by S. J. Lomonaco, Jr. A survey of quantum complexity theory by U. V. Vazirani Quantum error correcting codes and quantum cryptography: An introduction to quantum error correction by D. Gottesman A talk on quantum cryptography or how Alice outwits Eve by S. J. Lomonaco, Jr. More mathematical connections: Topological quantum codes and anyons by A. Kitaev Quantum topology and quantum computing by L. H. Kauffman An entangled tale of quantum entanglement by S. J. Lomonaco, Jr. Index.

These notes are intended as a simple introduction to the new field of quantum computing, quantum information theory and quantum cryptography.Undergraduate level quantum mechanics and mathematics is required for an understanding of these lectures.After an introduction to qubits and quantum registers, we introduce the key topics of entangled states and quantum logic gates.For two qubit states, we introduce the four Bell states as a change of basis.The essentials of quantum cryptography are then described, although this is just a straightforward application of quantum mechanics.The characters of Alice, Bob and Eve are first introduced here.Two qubit Bell states are used to demonstrate a novel 'dense coding' technique.Finally, in these communication applications, quantum teleportation is explained in detail, again making use of entangled Bell states.The technique of magnetic spin resonance is used as a familiar example to illustrate how qubit operations could in principle be realised.This leads on to the specification of quantum devices that can encode functions.All this is preparatory to a detailed discussion of two of the most significant quantum algorithms discovered to date, namely, Peter Shor's factorization algorithm and Lov Grover's quantum database search algorithm.

This is a set of lecture notes suitable for a Master's course on quantum computation and information from the perspective of theoretical computer science. The first version was written in 2011, with many extensions and improvements in subsequent years. The first 10 chapters cover the circuit model and the main quantum algorithms (Deutsch-Jozsa, Simon, Shor, Hidden Subgroup Problem, Grover, quantum walks, Hamiltonian simulation and HHL). They are followed by 4 chapters about complexity, 4 chapters about distributed ("Alice and Bob") settings, a chapter about quantum machine learning, one about stabilizer states and Clifford circuits, and a final chapter about quantum error correction. Appendices A and B give a brief introduction to the required linear algebra and some other mathematical and computer science background. All chapters come with exercises, with some hints provided in Appendix C.

Quantum physics allows entirely new forms of computation and cryptography, which could perform tasks currently impossible on classical devices, leading to an explosion of new algorithms, communications protocols and suggestions for physical implementations of all these ideas. As a result, quantum information has made the transition from an exotic research topic to part of mainstream undergraduate courses in physics. Based on years of teaching experience, this textbook builds from simple fundamental concepts to cover the essentials of the field. Aimed at physics undergraduate students with a basic background in quantum mechanics, it guides readers through theory and experiment, introducing all the central concepts without getting caught up in details. Worked examples and exercises make this useful as a self-study text for those who want a brief introduction before starting on more advanced books. Solutions are available online at www.cambridge.org/9781107014466.

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This paper gives an overview from the perspective of Lie group theory of some of the recent advances in the rapidly expanding research area of quantum entanglement. This paper is a written version of the last of eight one hour lectures given in the American Mathematical Society (AMS) Short Course on Quantum Computation held in conjunction with the Annual Meeting of the AMS in Washington, DC, USA in January 2000. More information about the AMS Short Course can be found at http://www.csee.umbc.edu/~lomonaco/ams/Announce.html