Could a quantum computer perform linear algebra faster than a classical computer? Supposing we had a quantum computer with a sufficient number of qubits, could we use it to do linear algebra faster than we could with a classical computer? What sort of speedup could we expect? Has anyone created a quantum algorithm for linear algebra, and what is it's running time? In theory, an operation such as matrix-matrix multiplication is highly parallelizable, however in practice it requires a lot of work to implement parallel matrix-matrix multiplication that runs quickly. Would a quantum computer provide any practical advantage?
Here are some pointers:


Quantum algorithm for linear systems of equations by Harrow, Hassidim, and Lloyd. This paper shows how to solve sparse systems of linear equations very quickly.
Quantum Algorithms for Linear Algebra and Machine Learning by Anupam Prakash. This PhD thesis proposes a quick algorithm for singular value estimation, and presents several applications.