Quantum Computation, Theory Of

1-The study of the model of computation in which the state space consists of linear superpositions of classical configurations and the computational steps consist of applying local unitary operators and measurements as permitted by quantum mechanics. Quantum computation emerged in the 1980's when P. Benioff and R. Feynman realized that the apparent exponential complexity in simulating quantum physics could be overcome by using a sufficiently well controlled quantum mechanical system to perform a simulation. Quantum Turing machines were introduced by D. Deutsch in 1985. Initial work focused on how quantum mechanics could be used to implement classical computation (computation in the sense of A. Church and A. Turing), and on analyzing whether the quantum Turing machine model provided a universal model of computation. In the early 1990's, D. Deutsch and R. Jozsa found an oracle problem that could be solved faster on an error-free quantum computer than on any deterministic classical computer. E. Bernstein and U. Vazirani then formalized the notion of quantum complexity from a theoretical computer science point of view, and showed that with respect to oracles which reversibly compute classical functions, quantum computers are super-polynomially more efficient than classical computers. The gap was soon improved to an exponential one. This work culminated in P. Shor's discovery of an efficient (that is, consuming only polynomial resources) algorithm for factoring large numbers and for computing discrete logarithms. It implied that widely used public key cryptographic systems would be insecure if quantum computers were available. Subsequently, L. Grover found an algorithm which permitted a square-root speed-up of unstructured search. Finding new algorithmic improvements achievable with quantum computers which are not reducible to Shor's or Grover's algorithm is currently (2000) an active research area. Also of great current interest is understanding how the problem of simulating quantum systems, known to be tractable on a quantum computer, relates to the problems conventionally studied within classical computational complexity theory. Comprehensive introductions to quantum computation and the known quantum algorithms may be found in [4, 3]. The algorithmic work described above firmly established the field of quantum computation in computer science. However, it was initially unclear whether quantum computation was a physically realizable model. Particularly worrisome was the fact that in nature, quantum effects are