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arXiv:2212.03939

Optimal Quantum Algorithm for Vector Interpolation

Sophie M. Decoppet

Published Dec 7, 2022• 0 citations
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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.

DOI
10.48550/arxiv.2212.03939

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How many quantum queries are required to determine the coefficients of a degree- d polynomial in n variables? We present and analyse quantum algorithms for this multivariate polynomial interpolation problem over the fields F q , R and C . We show that k C and 2 k C queries suffice to achieve probability 1 for C and R , respectively, where k C = ⌈ ( 1 / ( n + 1 ) ) ( n + d d ) ⌉ except for d =2 and four other special cases. For F q , we show that ⌈( d /( n + d ))( n + d d ) ⌉ queries suffice to achieve probability approaching 1 for large field order q . The classical query complexity of this problem is ( n + d d ) , so our result provides a speed-up by a factor of n +1, ( n +1)/2 and ( n + d )/ d for C , R and F q
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