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In this paper we present a canonical quantum computing method to estimate theweighted sum w(k)f(k) of the values taken by a discrete function f and realweights w(k). The canonical aspect of the method comes from relying on a singlelinear function encoded in the amplitudes of a quantum state, and usingregister entangling to encode the function f.  We further expand this framework by mapping function values to hashes inorder to estimate weighted sums w(k)h(f(k)) of hashed function values with realhashes h. This generalization allows the computation of restricted weightedsums such as value at risk, comparators, as well as Lebesgue integrals andpartial moments of statistical distributions.  We also introduce essential building blocks such as efficient encodings ofstandardized linear quantum states and normal distributions.

A Generalized Quantum Inner Product and Applications to Financial Engineering