Samuel L. Braunstein
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 . 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 . 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 . 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 . 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.