Optimal Quantum Algorithm for Vector Interpolation

• URL: http://arxiv.org/abs/2212.03939v1
• Date: Wed, 7 Dec 2022 20:08:24 GMT
• Title: Optimal Quantum Algorithm for Vector Interpolation
• Authors: Sophie Decoppet
• Abstract summary: We study the functions that can be learned through the functions that can be learned through the In quantum algorithm designed by Childs et al. We show that the success probability approaches 1 for large $q$ and large domain order $|mathcalV|. We provide a conservative formula for the number of queries required to achieve this success probability.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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