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A quantum computer operates on
a quantum state, rather than
a classical one.
One reason this is interesting
is because of the possibility
of using all this
additional richness
of these entangled states.
The state of a classical
machine with N objects
that can each be in one
of two different states
can be described
by N binary digits.
By contrast, a machine dealing
with N two-level quantum
mechanical systems could be
performing an operation on two
to the power N numbers at once.
No classical machine can do
that, even for moderate N.
As we said, for N equal to 300
quantum mechanical objects,
each with two states, we need
two to the power 300 objects
to store the information for
the quantum mechanical state,
and there simply are not
enough atoms to store two
to the power 300 numbers
at one atom per number.
It's well known in
computer science
that many problems have
the unfortunate property
that the required
complexity grows very fast
as the scale or the size
of the problem increases.
No classical computer
could possibly
solve such problems once they
get above a certain size.
Perhaps a quantum
computer working
with quantum mechanical
systems of some moderate size
could solve such problems.
Quantum computer
algorithms have been
developed for at least
two important problems
of this type.
The first such quantum
algorithm to be proposed
was for finding factors of
large numbers, a notoriously
hard computational problem.
This problem is so
hard that we routinely
use that hardness in
constructing the encryption
algorithms we use every day
to send secure information.
The algorithm that would
run on a quantum computer
to factorize large numbers
is called the Shor algorithm,
after its inventor, Peter Shor.
The second such
quantum algorithm
could be used for
searching databases,
a problem essentially like
trying to look up a telephone
directory backwards.
Given the telephone
number, can we
figure out whose number it is?
This is known as the
Grover algorithm,
after its inventor, Lov Grover.
A quantum computer is, however,
not necessarily an easy thing
to make.
One thing we should
understand is
that it would work in a
very different way compared
to any classical
digital computer.
A classical computer
takes data as bits,
processes it in some
kind of black box,
and gives an output as bits.
A quantum computer,
however, would not
work with bits in the
conventional sense.
Instead of bits that
are only zero or one,
it would work with
what are called qubits.
"Qubit" is spelled Q-U-B-I-T.
Qubits are represented
as the linear superposition
of two states of a quantum
mechanical object.
The object might be
an electron's spin
in the linear superposition
of spin up and spin down,
or it might be a photon in a
linear superposition of two
different polarization
states, or it
might be an atom in a
linear superposition of two
possible states, a
lower, or ground, state,
and an upper, or excited, state.
So we could write a qubit
state in the following form.
We could call the
state psi, if we like,
and write it like this.
And here, zero would be the
quantum mechanical state
that is representing zero,
or logic zero, if you like.
For example, it could be a
horizontal polarization state
H of a photon, or
perhaps it could
be a spin-down state
of an electron,
or possibly a ground state--
g, for example-- of an atom.
And similarly, in
this expression,
one could be the quantum
mechanical state representing
our logic one, if you like, and
could be physically represented
by, for example, a vertical
polarization, or spin-up,
or an excited atomic
state, maybe e.
Because of normalization,
these two coefficients
have to be such that the sum
of their modulus squareds
adds up to one.
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