Quantum Computing and Entanglement for Mathematicians

Abstract

These notes are an expanded form of lectures presented at the C.I.M.E. summer school in representation theory in Venice, June 2004. The sections of this article roughly follow the five lectures given. The first three lectures (sections) are meant to give an introduction to an audience of mathematicians (or mathematics graduate students) to quantum computing. No attempt is given to describe an implementation of a quantum computer (it is still not absolutely clear that any exist). There are also some simplifying assumptions that have been made in these lectures. The short introduction to quantum mechanics in the first section involves an interpretation of measurement that is still being debated which involves the "collapse of the wave function" after a measurement. This interpretation is not absolutely necessary but it simplifies the discussion of quantum error correction. The next two sections give an introduction to quantum algorithms and error correction through examples including fairly complete explanations of Grover's (unordered search) and Shor's (period search and factorization) algorithms and the quantum perfect (five qubit) code. The last two sections present applications of representation and Lie theory to the subject. We have emphasized the applications to entanglement since this is the most mathematical part of recent research in the field and this is also the main area to which the author has made contributions. The material in subsections 5.1 and 5.3 appears in this article for the first time.