An Introduction to Quantum Computing Algorithms

In 1994 Peter Shor [65] published a factoring algorithm for a quantum computer that finds the prime factors of a composite integer N more efficiently than is possible with the known algorithms for a classical com puter. Since the difficulty of the factoring problem is crucial for the se curity of a public key encryption system, interest (and funding) in quan tum computing and quantum computation suddenly blossomed. Quan tum computing had arrived. The study of the role of quantum mechanics in the theory of computa tion seems to have begun in the early 1980s with the publications of Paul Benioff [6]' [7] who considered a quantum mechanical model of computers and the computation process. A related question was discussed shortly thereafter by Richard Feynman [35] who began from a different perspec tive by asking what kind of computer should be used to simulate physics. His analysis led him to the belief that with a suitable class of "quantum machines" one could imitate any quantum system.

[see attached for complete text]Over the last decade quantum computing has become a rapidlygrowing field with relevance to topics from physics, mathematics, andcomputer science. Aimed at engaging the mathematically literatereader who may not have a background in all of the subspecialties ofthe field, this work will provide the context for understanding anddeveloping algorithms for a "prospective" quantum computer.Key features include:* clear and concise exposition, * minimal prerequisites, * detailedoverview of the historical context of quantum computing, * discussionof the most recent developments, including the factoring algorithm of

1 Quantum.- 2 Basics of Quantum Computation.- 3 Quantum Algorithms.- 4 Quantum Error-Correcting Codes.- Afterword.- References.