Quantum Computing in Complexity Theory and Theory of Computation

Traditionally, complexity theory has concerned itself with algorithms run by classical computers. With the recent developments in quantum computing, complexity theorists have begun considering just how quantum algorithms fit into the picture. After a brief introduction into classical complexity theory, we consider what is known and what is speculated about the relationship of the class of problems solvable by quantum algorithms (denoted BQP) and the well-known classical classes P and NP. A short discussion on the possibility of hypercomputation (that is, the ability to solve problems which are undecidable on a classical Turing machine) follows.