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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. 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Childs, Jianxin Chen, Shih-Han Hung","submitted_at":"2017-01-15T04:07:40Z","abstract_excerpt":"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. 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