  A central problem in quantum computing is to identify computational tasks
which can be solved substantially faster on a quantum computer than on any
classical computer. By studying the hardest such tasks, known as BQP-complete
problems, we deepen our understanding of the power and limitations of quantum
computers. We present several BQP-complete problems, including Local
Hamiltonian Eigenvalue Sampling and Phase Estimation Sampling. Different than
the previous known BQP-complete problems (the Quadratically Signed Weight
Enumerator problem [KL01] and the Approximation of Jones Polynomials [FKW02,
FLW02, AJL06]), our problems are of a basic linear algebra nature and are
closely related to the well-known quantum algorithm and quantum complexity
theories.
