Answer 1:
The idea of quantum computers is very
interesting, but also very complex as it is rooted
in the fundamental ideas of quantum
mechanics. The "classical" computers that we
are all familiar with are based on binary bits.
This means that each bit can be in one of two
different states, 0 or 1. Another feature is that
classical computers can only do calculations on
one set of numbers at a time. Quantum computers
on the other hand, are based on quantum bits, or
"qubits." Whereas a binary bit can only be a
0 or a 1, a qubit can be a 0, a 1, or some
combination, or superposition, of the two states.
Qubits are what would make quantum computers
so powerful. To illustrate, let's imagine a system
comprised of some number, n, different bits. In
terms of binary bits, each bit would either be a 0
or a 1 and there would be 2^{n} different
ways that these could appear. For example, if
n=2, there are 2^{2} or 4 possibilities 
you could have 0,0 or 0,1 or 1,0 or 1,1. You can
see how one might have to do (up to) 4 different
calculations (one for each possible combination)
to cover all the possibilities. Now, if these
bits are instead qubits, each bit exists as a
superposition of the two possible states, meaning
that all four of the possibilities can be
considered at once. Hence we can effectively do
an operation on all the combinations at once! It
is this inherent quantum parallelism in the
computing that would make a quantum computer so
powerful. Furthermore, the advantages become more
clear as you increase n. For n=100, there are
2^{100} (~10^{30}) possibilities
of 0s and 1s! I think I have read that a quantum
computer with as few as 25 to 30 qubits would be
comparable for some types of calculations to the
most powerful supercomputers of today.
One other thing to think about is how to get
the answer you want amidst all those
possibilities? Earlier, we said that qubits
exist as a superposition of possible states.
Here, it might be useful to think of the
superposition of states as a (probability) wave.
If we think of each qubit as a wave, then we can
imagine how those waves can interfere with each
other (in the same way that waves on water
interact with each other). Thus, the idea is that
depending on what conditions are set on the
qubits, the right answer will appear from
constructive interference, and all the other
answers will be eliminated by the destructive
interference. Here, I should note that quantum
computers are many years away from becoming a
"practical" item as much research is currently
focused on making and controlling qubits, and
other problems such as decoherence.
Furthermore, quantum computers may not be
perfectly suited for every type of calculation.
However, one calculation it would be extremely
proficient at (leading to much of the funding for
research in this field) would be executing
algorithms to factor very large numbers. The
reason that is so significant is because many
encryption codes (inclusing RSA) depend on the
difficulty of factoring very large numbers, and a
quantum computer could somewhat easily crack these
codes.
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