Prime Numbers

8.5 Quantum computation 419
We speak of quantum computation, which is to be thought of as a genuine
replacement for computer processes as we have previously understood them.
The first basic notion is a distinction between classical Turing machines (TMs)
and quantum Turing machines (QTMs). The older TM model is the model
of every prevailing computer of today, with the possible exception of very
minuscule, tentative and experimental QTMs, in the form of small atomic
experiments and so on. (Although one could argue that nature has been
running a massive QTM for billions of years.) The primary feature of a TM is
that it processes “serially,” in following a recipe of instructions (a program)
in a deterministic fashion. (There is such a notion as a probabilistic TM
behaving statistically, but we wish to simplify this overview and will avoid
that conceptual pathway.) On the other hand, a QTM would be a device in
which a certain “parallelism” of nature would be used to effect computations
with truly unprecedented efficiency. That parallelism is, of course, nature’s
way of behaving according to laws of quantum mechanics. These laws involve
many counterintuitive concepts. As students of quantum theory know, the
microscopic phenomena in question do not occur as in the macroscopic world.
There is the particle–wave duality (is an electron a wave or a particle or
both?), the notion of amplitudes, probability, interference—not just among
waves but among actual parcels of matter—and so on. The next section is a
very brief outline of quantum computation concepts, intended to convey some
qualitative features of this brand new science.
8.5.1 Intuition on quantum Turing machines (QTMs)
Because QTMs are still overwhelmingly experimental, not having solved a
single “useful” problem so far, we think it appropriate to sketch, mainly
by analogy, what kind of behavior could be expected from a QTM. Think
of holography, that science whereby a solid three-dimensional object is cast
onto a planar “hologram.” What nature does is actually to “evaluate” a 3dimensional
Fourier transform whose local power fluctuations determine what
is actually developed on the hologram. Because light moves about one foot
in a nanosecond (10 −9 seconds), one can legitimately say that when a laser
light beam strikes an object (say a chess piece) and the reflections are mixed
with a reference beam to generate a hologram, “nature performed a huge
FFT in a couple of nanoseconds.” In a qualitative but striking sense, a known
O(N ln N) algorithm (where N would be sufficiently many discrete spatial
points to render a high-fidelity hologram, say) has turned into more like an
O(1) one. Though it is somewhat facetious to employ our big-O notation in
this context, we wish only to make the point that there is parallelism in the
light-wave-interference model that underlies holography. On the film plane of
the hologram, the final light intensity depends on every point on the chess
piece. This is the holographic, one could say “parallel,” aspect. And QTM
proposals are reminiscent of this effect.
We are not saying that a laboratory hologram setup is a QTM, for some
ingredients are missing in that simplistic scenario. For one thing, modern QTM