Prime Numbers

420 Chapter 8 THE UBIQUITY OF PRIME NUMBERS
theory has two other important elements beyond the principle of quantum
interference; namely, probabilistic behavior, and a theoretical foundation
involving operators such as unitary matrices. For another thing, we would like
any practical QTM to bear not just on optical experiments, but also on some
of the very difficult tasks faced by standard TMs—tasks such as the factoring
of large integers. As have been a great many new ideas, the QTM notion
was pioneered in large measure by the eminent R. Feynman, who observed
that quantum-mechanical model calculations tend, on a conventional TM, to
suffer an exponential slowdown. Feynman even devised an explicit model of
a QTM based on individual quantum registers [Feynman 1982, 1985]. The
first formal definition was provided by [Deutsch 1982, 1985], to which current
formal treatments more or less adhere. An excellent treatment—which sits
conveniently between a lay perspective and a mathematical one—is [Williams
and Clearwater 1998]. On the more technical side of the physics, and some
of the relevant number-theoretical ideas, a good reference is [Ekert and Jozsa
1996]. For a very accessible lay treatment of quantum computation, see [Hey
1999], and for course-level material see [Preskill 1999].
Let us add a little more quantum flavor to the idea of laser light calculating
an FFT, nature’s way. There is in quantum theory an ideal system called the
quantum oscillator. Given a potential function V (x) =x 2 , the Schrödinger
equation amounts to a prescription for how a wave packet ψ(x, t), where t
denotes time, moves under the potential’s influence. The classical analogue
is a simple mass-on-a-spring system, giving smooth oscillations of period τ,
say. The quantum model also has oscillations, but they exhibit the following
striking phenomenon: After one quarter of the classical period τ, an initial
wave packet evolves into its own Fourier transform. This suggests that you
could somehow load data into a QTM as an initial function ψ(x, 0), and later
read off ψ(x, τ/4) as an FFT. (Incidentally, this idea underlies the discussion
around the Riemann-ζ representation (8.5).) What we are saying is that the
laser hologram scenario has an analogue involving particles and dynamics.
We note also that wave functions ψ are complex amplitudes, with |ψ| 2 being
probability density, so this is how statistical features of quantum theory enter
into the picture.
Moving now somewhat more toward the quantitative, and to prepare for
the rest of this section, we presently lay down a few specific QTM concepts.
It is important right at the outset, especially when number-theoretical
algorithms are involved, to realize that an exponential number of quantities
may be “polynomially stored” on a QTM. For example, here is how we can
store in some fashion—in a so-called quantum register—every integer a ∈
[0,q− 1],inonlylgq so-called qbits. At first this seems impossible, but recall
our admission that the quantum world can be notoriously counterintuitive.
A mental picture will help here. Let q =2 d , so that we shall construct a
quantum register having d qbits. Now imagine a line of d individual ammonia
molecules, each molecule being NH3 in chemical notation, thought of as a
tetrahedron formed by the three hydrogens and a nitrogen apex. The N apex
is to be thought of as “up” or “down,” 1 or 0, i.e., either above or below the