In this paper we give a tutorial on 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']) was successfully
factored using this algorithm on approximately 1600 workstations scattered
around the world; the entire factorization took eight months
[4]. Using
this to estimate the prefactor of the above exponential scaling, we
find that it would take roughly 800,000 years to factor a 250 digit
number with the same computer power; similarly, a 1000 digit number
would require 
years
(significantly lon ger than the age of
the universe). The difficulty of factoring large numbers is crucial
for public-key cryptosystems, such as ones used by banks. There, such
codes rely on the difficulty of factoring numbers with around 250
digits.
Recently, an algorithm was developed for factoring numbers on a quantum
computer which runs in 
steps where
is small [1]. This is roughly quadratic in the
input size, so factoring a 1000 digit number with such an
algorithm would require only a few million steps. The implication is
that public key cryptosystems based on factoring may be breakable.
To give you an idea of how this exponential improvement might be possible, we review an elementary quantum mechanical experiment that demonstrates where such power may lie hidden [5]. The two-slit experiment is prototypic for observing quantum mechanical behavior: A source emits photons, electrons or other particles that arrive at a pair of slits. These particles undergo unitary evolution and finally measurement. We see an interference pattern, with both slits open, which wholely vanishes if either slit is covered. In some sense, the particles pass through both slits in parallel. If such unitary evolution were to represent a calculation (or an operation within a calculation) then the quantum system would be performing computations in parallel. Quantum parallelism comes for free. The output of this system would be given by the constructive interference among the parallel computations.