This book is a collection of papers given by invited speakers at the AMS Special Session on Quantum Computation and Information held at the January 2000 Annual Meeting of the AMS in Washington, DC. The papers in this volume give readers a broad introduction to the many mathematical research challenges posed by the new and emerging field of quantum computation and quantum information. Of particular interest is a long paper by Lomonaco and Kauffman discussing mathematical and computational aspects of the so-called hidden subgroup algorithm.This book is intended to help readers recognize that, as a result of this new field of quantum information science, mathematical research opportunities abound in such diverse mathematical fields as algebraic coding theory, algebraic geometry, algebraic topology, communication theory, control theory, cryptography, differential geometry, differential topology, dynamical systems, game theory, group theory, information theory, number theory, operator theory, robotics, theory of computation, mathematical logic, mathematical physics, and more.It is hoped that this book will act as a catalyst to encourage members of the mathematical community to take advantage of the many mathematical research opportunities arising from the 'grand challenge' of Quantum Information Science. This book is the companion volume to ""Quantum Computation: A Grand Mathematical Challenge for the Twenty-First Century and the Millennium"", PSAPM/58, Volume 58 in the ""Proceedings of Symposia in Applied Mathematics"" series.