# Quantum algorithm for multivariate polynomial interpolation

## TL;DR

This paper develops quantum algorithms for multivariate polynomial interpolation over various fields, significantly reducing the number of queries needed compared to classical methods, and highlights a larger quantum speedup in multivariate cases.

## Contribution

The paper introduces quantum algorithms that achieve polynomial interpolation with fewer queries than classical algorithms over different fields, expanding the known quantum advantage.

## Key findings

- Quantum algorithms require fewer queries than classical methods.
- Achieves near-certain interpolation with fewer than classical queries.
- Demonstrates larger quantum speedups in multivariate cases than univariate cases.

## Abstract

How many quantum queries are required to determine the coefficients of a degree-$d$ polynomial in $n$ variables? We present and analyze quantum algorithms for this multivariate polynomial interpolation problem over the fields $\mathbb{F}_q$, $\mathbb{R}$, and $\mathbb{C}$. We show that $k_{\mathbb{C}}$ and $2k_{\mathbb{C}}$ queries suffice to achieve probability $1$ for $\mathbb{C}$ and $\mathbb{R}$, respectively, where $k_{\mathbb{C}}=\smash{\lceil\frac{1}{n+1}{n+d\choose d}\rceil}$ except for $d=2$ and four other special cases. For $\mathbb{F}_q$, we show that $\smash{\lceil\frac{d}{n+d}{n+d\choose d}\rceil}$ queries suffice to achieve probability approaching $1$ for large field order $q$. The classical query complexity of this problem is $\smash{n+d\choose d}$, so our result provides a speedup by a factor of $n+1$, $\frac{n+1}{2}$, and $\frac{n+d}{d}$ for $\mathbb{C}$, $\mathbb{R}$, and $\mathbb{F}_q$, respectively. Thus we find a much larger gap between classical and quantum algorithms than the univariate case, where the speedup is by a factor of $2$. For the case of $\mathbb{F}_q$, we conjecture that $2k_{\mathbb{C}}$ queries also suffice to achieve probability approaching $1$ for large field order $q$, although we leave this as an open problem.