Quantum matrix multiplication?
In arXiv:quant-ph/0409035v2 Buhrman and Spalek present a quantum algorithm beating the Coppersmith-Winograd algorithm in cases where the output matrix has few nonzero entries.

Update: There is also a slightly improved quantum algorithm by Dörn and Thierauf.

Update: There is an improved quantum algorithm by Le Gall beating Burhman and Spalek in general.