Quantum algorithm for multivariate polynomial interpolation

Abstract

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 Fq, R and C. We show that kC and 2kC queries suffice to achieve probability 1 for C and R, respectively, where kC = (1/(n + 1))n+dd except for d = 2 and four other special cases. For Fq, we show that (d/(n + d))n+dd queries suffice to achieve probability approaching 1 for large field order q. The classical query complexity of this problem is n+dd, so our result provides a speed-up 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.