A list of the efficient (a lot better than algorithms for regular computers)
Shor's algorithm, an algorithm for factoring numbers, which is key to decryption
Solving pell's equation Pell's equation is useful for approximating things like the square root of 2
Estimating certain Gauss sums
Solving hidden shift problems
solving certain hidden subgroup problems
Every quantum algorithm is fourier transforms and classical computation.
The Toffoli gate and the Hadamard gate are universal for quantum computing
the Toffoli gate, invented by Tommaso Toffoli, is a universal reversible logic gate, which means that any reversible circuit can be constructed from Toffoli gates. It is also known as the "controlled-controlled-not" gate, which also describes its action.