Is there any proof that quantum computers are more efficient than classical computers?

Shor's algorithm is often used as the argument. It can solve the factorization problem faster than any known algorithm for classical computers. Yet, we have no proof classical computers can't also factor integers efficiently.

Is there any actual proof quantum computers can solve some problems faster than classical computers?

Yes, Grover's algorithm shows you can use a quantum algorithm to find an element in an unordered database of size $N$ with high probability by querying the database only $O(\sqrt{N})$ times. Any classical solution that succeeds with high probability requires $\Omega (N)$ queries to the database.

StackExchange Computer Science Q#50366, answer score: 18