Prime Numbers

8.5 Quantum computation 421
three H’s. Thus, any d-bit binary number can be represented by a collective
orientation of the molecules. But what about representing all possible binary
strings of length d? This turns out to be easy, because of a remarkable quantum
property: An ammonia molecule can be in both 1, 0 states at the same time.
One way to think of this is that lowest-energy states—called ground states—
are symmetrical when the geometry is. A container of ammonia in its ground
state has each molecule somehow “halfway present” at each 0, 1 position.
In theoretical notation we say that the ground state of one ammonia qbit
(molecule, in this model) is given by:
φ = 1
√ 2 ( | 0 〉 + | 1 〉 ),
where the “bra-ket” notation |〉is standard (see the aforementioned quantumtheoretical
references). The notation reminds us that a state belongs to
an abstract Hilbert space, and only an inner product can bring this back
to a measurable number. For example, given the ground state φ here, the
probability that we find the molecule in state | 0 〉 is the squared inner product
|〈0 | φ 〉| 2
=
1
√ 2 〈 0 | 0 〉
2
= 1
2 ,
i.e., 50 per cent chance that the nitrogen atom is measured to be “down.” Now
back to the whole quantum register of d qbits (molecules). If each molecule is
in the ground state φ, then in some sense every single d-bit binary string is
represented. In fact, we can describe the state of the entire register as [Shor
1999]
ψ = 1
2d/2 2 d −1
| a 〉,
where now |a〉 denotes the composite state given by the molecular orientations
corresponding to the binary bits of a; for example, for d = 5 the state |10110〉
is the state in which the nitrogens are oriented “up, down, up, up, down.” This
is not so magical as it sounds, when one realizes that now the probability of
finding the entire register in a particular state a ∈ [0, 2 d − 1] is just 1/2 d .It
is this sense in which every integer a is stored—the collection of all a values
is a “superposition” in the register.
Given a state that involves every integer a ∈ [0,q − 1], we can imagine
acting on the qbits with unitary operators. For example, we might alter the
0-th and 7-th qbits by acting on the two states with a matrix operator.
An immediate physical analogy here would be the processing of two input
light beams, each possibly polarized up or down, via some slit interference
experiment (having polaroid filters within) in which two beams are output.
Such a unitary transformation preserves overall probabilities by redistributing
amplitudes between states.
a=0