Optimal Quantum Algorithm for Vector Interpolation

TL;DR

This paper extends a quantum polynomial interpolation algorithm to efficiently learn vector inner product functions over finite fields, proving its optimality and analyzing success probabilities and query complexity.

Contribution

It broadens the scope of the polynomial interpolation quantum algorithm to vector inner product functions and establishes its optimality and success probability bounds.

Findings

Algorithm is optimal for vector inner product functions.

Success probability approaches 1 for large q and domain size.

Provides a conservative query complexity formula.

Abstract

In this paper we study the functions that can be learned through the polynomial interpolation quantum algorithm designed by Childs et al. This algorithm was initially intended to find the coefficients of a multivariate polynomial function defined on finite fields $\mathbb{F}_q$. We extend its scope to vector inner product functions of the form $\mathcal{O}_{\mathbf{s}}(\mathbf{v}) = \mathbf{s}\cdot\mathbf{v}$ where the goal is to find the vector $\mathbf{s} \in \mathbb{F}_q^n$. We examine the necessary conditions on the domain $\mathcal{V}$ of $\mathcal{O}_{\mathbf{s}}$ and prove that the algorithm is optimal for such functions. Furthermore, we show that the success probability approaches 1 for large $q$ and large domain order $|\mathcal{V}|.$ Finally, we provide a conservative formula for the number of queries required to achieve this success probability.