Quantum Computation Computation Is a Physical Process !

As computers become faster they must become smaller because of the niteness of the speed of light. The history of computer technology has involved a sequence of changes from one type of physical realisation to another-from gears to relays to valves to transistors to integrated circuits and so on. Quantum mechanics is already important in the design of microelectronic components. Soon it will be necessary to harness quantum mechanics rather than simply take it into account, and at that point it will be possible to give data processing devices new functionality. Computers are physical objects, and computations are physical processes. This sentence, innocuous at rst glance, leads to non-trivial consequences, some of which we wish to explore in this paper. In particular we want to convince the reader that the theory of computation is not a branch of pure mathematics. Fundamental questions regarding computability and computational complexity are questions about physical processes that reveal to us properties of abstract entities such as numbers or ideas. Those questions belong to physics rather than mathematics (1,2). For the purpose of this presentation we deene a computation as a physical process that produces nal states, outputs, that depend in some desired way on given initial states, inputs. Quantum computers are deened as physical devices whose unitary dynamics can be regarded as the performance of a computation. Unlike classical computers, quantum computers can operate on quantum superpositions of diierent numbers. In recent years a new quantum theory of computation has been developed (3{8). It is in many ways diierent from the classical, purely mathematical theory of computation. A single quantum computer can follow many distinct computation paths all at the same time and produce a nal output depending on the interference of all of them. In particular, it has been shown recently that quantum computers can eeciently perform classes of computation , e.g. factorisation, which are believed to be intractable on any classical computer (9). This makes it highly desirable to construct such devices. This