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Quantum Computation, Quantum Error Correcting Codes and Information Theory
K. R. Parthasarathy, Indian Statistical Institute, New Delhi, India
A publication of the Tata Institute of Fundamental Research.
cover
Tata Institute of Fundamental Research
2005; 120 pp; softcover
ISBN-10: 81-7319-688-5
ISBN-13: 978-81-7319-688-1
List Price: US$20
Member Price: US$16
Order Code: TIFR/7
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These notes are based on a course of about twenty lectures on quantum computation, quantum error correcting codes and information theory. The topics include a comparative description of the basic features of classical probability theory on finite sample spaces and quantum probability theory on finite dimensional complex Hilbert spaces, quantum gates and cicuits, simple examples of circuits arising from quantum teleportation, communication through EPR pairs and arithmetical computations on a quantum computer, more sophisticated examples of such circuits in the context of Fourier transform and phase estimation, a detailed account of the order finding algorithm as well as the celebrated Shor's algorithm for factorising a positive integer into its prime factors.

There is a leisurely discussion of quantum error correcting codes with the Knill-Laflamme criterion for error correction and a number of examples of such codes whose construction is based on the Weyl commutation relations for finite abelian groups. The reader may find here a brief introduction to the basic ideas of classical information theory as developed by Shannon,properties of von Neumann's quantum entropy and relative entropy as well as a proof of Schumacher's noiseless quantum coding theorem. The Holevo bound for transmission of classical information through encoding by quantum states followed by measurements is derived.

The only background assumed of the reader is linear algebra on finite dimensional complex vector spaces and elementary classical probability theory on finite sample spaces.These notes are aimed at mathematicians and computer scientists who are curious to know the "mystery" behind a quantum computer and the possibility of communicating information using the principles of elementary quantum theory.

A publication of the Tata Institute of Fundamental Research. Distributed worldwide except in India, Bangladesh, Bhutan, Maldavis, Nepal, Pakistan, and Sri Lanka.

Readership

Graduate students, research mathematicians, and computer scientists interested in quantum computing.

Table of Contents

  • Quantum probability
  • Quantum gates and circuits
  • Universal quantum gates
  • The Fourier transform and an application
  • Order finding
  • Shor's algorithm
  • Quantum error correcting codes
  • Classical information theory
  • Quantum information theory
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