gral part of the world view of anyone who seeks a fundamental understanding of the quantum theory and the processing of information. How quantum mechanics can be used to improve computation. Our challenge: solving an exponentially difficult problem for a conventional computer---that of factoring a large number. As a prelude, we review the standard tools of computation, universal gates and machines. These ideas are then applied first to classical, dissipationless computers and then to quantum computers. A schematic model of a quantum computer is described as well as some of the subtleties in its programming. The Shor algorithm [1,2] for efficiently factoring numbers on a quantum computer is presented in two parts: the quantum procedure within the algorithm and the classical algorithm that calls the quantum procedure. The mathematical structure in factoring which makes the Shor algorithm possible is discussed. We conclude with an outlook to the feasibility and prospects for quantum computation in the coming years. Let us start by describing the problem at hand: factoring a number N into its prime factors (e.g., the number 51688 may be decomposed as ). A convenient way to quantify how quickly a particular algorithm may solve a problem is to ask how the number of steps to complete the algorithm scales with the size of the ``input'' the algorithm is fed. For the factoring problem, this input is just the number N we wish to factor; hence the length of the input is . (The base of the logarithm is determined by our numbering system. Thus a base of 2 gives the length in binary; a base of 10 in decimal.) `Reasonable' algorithms are ones which scale as some small-degree polynomial in the input size (with a degree of perhaps 2 or 3). On conventional computers the best known factoring algorithm runs in steps [3]. This algorithm, therefore, scales exponentially with the input size . For instance, in 1994 a 129 digit number (known as RSA129 [3']) ...
