Quantum algorithm for multivariate polynomial interpolation

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 [Formula: see text], [Formula: see text] and [Formula: see text]. We show that [Formula: see text] and [Formula: see text] queries suffice to achieve probability 1 for [Formula: see text] and [Formula: see text], respectively, where [Formula: see text] except for d=2 and four other special cases. For [Formula: see text], 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 [Formula: see text], [Formula: see text] and [Formula: see text], 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 [Formula: see text], we conjecture that [Formula: see text] queries also suffice to achieve probability approaching 1 for large field order q, although we leave this as an open problem.