Classical and Quantum Computers

William Seager wrote:

Quantum computers can perform calculations
faster than any classical computer it appears, but it is not
known that they can compute functions which Turing machines
cannot. In fact, there is no *proof* that quantum computers
are even faster than classical computers (since there is no
proof of p <> np). If we don't know that, we don't have a
proof that classical computers can't factor primes efficiently.
And of course, even if not efficiently, classical computers
can factor prime numbers ...

[for a great intro see M. Nielsen and I. Chuang *Quantum
Computation and Quantum Information*, Part 1].

(Alfredo) Nielsen and Chuang wrote (p. 6-7) that Shor's
demonstration of two problems - finding prime factors of
an integer and the 'discrete logarithm' one - "attracted
widespread interest because these two problems were
and still are widely believed to have no efficient solution
on a classical computer. Shor's results are a powerful
indication that quantum computers are more powerful
than Turing machines".
They write the same in p. 216, that some problems are
"infeasible" or "intractable" in classical computers,
but not in quantum computers.
You can say it is only a matter of speed, but if you
need infinite time to execute a function in a classical
computer, for practical purposes they don't execute it...

Alfredo Pereira Jr.