A Generalized Quantum Inner Product and Applications to Financial Engineering

TL;DR

This paper introduces a quantum computing framework for efficiently estimating weighted sums of functions, including complex financial metrics, by encoding functions and distributions into quantum states.

Contribution

It presents a generalized quantum method for computing weighted sums and integrals, expanding quantum applications in financial engineering and statistical analysis.

Findings

Efficient quantum encoding of linear states and distributions.

Framework for estimating complex financial metrics like value at risk.

Generalization to hashed function values for broader applications.

Abstract

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