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Next: Computing at the atomic scale

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QUANTUM COMPUTATION: A TUTORIAL COOL CENTRAL SITE OF THE HOUR

Samuel L. Braunstein

Abstract:

Imagine a computer whose memory is exponentially larger than its
apparent physical size; a computer that can manipulate an exponential
set of inputs simultaneously; a computer that computes in the twilight
zone of Hilbert space. You would be thinking of a quantum computer.
Relatively few and simple concepts from quantum mechanics are needed
to make quantum computers a possibility. The subtlety has been in
learning to manipulate these concepts. Is such a computer an
inevitability or will it be too difficult to build?

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
system would be given by the constructive interference among the
parallel computations.

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* Computing at the atomic scale:
* Reversible computation:
* Classical universal machines and logic gates:
* FANOUT and ERASE:
* Computation without ERASE:
* Elementary quantum notation:
* Logic gates for quantum bits:
* Model quantum computer and quantum code:
* Quantum parallelism: Period of a sequence:
* Factoring numbers:
* Prospects:
* Appendix:
* Acknowledgements:
* References

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Next: Computing at the

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Samuel L. Braunstein
Wed Aug 23 11:54:31 IDT 1995

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