# Quantum Computation, Quantum Error Correcting Codes and Information Theory

Share this page
*K. R. Parthasarathy*

A publication of the Tata Institute of Fundamental Research

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.