  We prove that quantum computation is polynomially equivalent to classical
probabilistic computation with an oracle for estimating the value of simple
sums, quadratically signed weight enumerators. The problem of estimating these
sums can be cast in terms of promise problems and has two interesting variants.
An oracle for the unconstrained variant may be more powerful than quantum
computation, while an oracle for a more constrained variant is efficiently
solvable in the one-bit model of quantum computation. Thus, problems involving
estimation of quadratically signed weight enumerators yield problems in BQP
(bounded error quantum polynomial time) that are distinct from the ones studied
so far, include a canonical BQP complete problem, and can be used to define and
study complexity classes and their relationships to quantum computation.
