



01497nas a2200145   4500008004100000245006400041210006400105260001500169490000800184520103000192100001801222700002301240700001901263856006901282       2018                        eng d00aQuantum algorithm for multivariate polynomial interpolation0 aQuantum algorithm for multivariate polynomial interpolation  c2018/01/170 v4743 a<p>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 Fq, R, and C. We show that kC and 2kC queries suffice to achieve probability 1 for C and R, respectively, where kC = &lceil; 1 n+1 ( n+d d )&rceil; except for d = 2 and four other special cases. For Fq, we show that &lceil; d n+d ( n+d d )&rceil; 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 speedup by a factor of n + 1, n+1 2 , and n+d d for C, R, and Fq, 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 Fq, we conjecture that 2kC queries also suffice to achieve probability approaching 1 for large field order q, although we leave this as an open problem.</p>1 aChen, Jianxin1 aChilds, Andrew, M.1 aHung, Shih-Han  uhttp://rspa.royalsocietypublishing.org/content/474/2209/20170480