Quantum algorithm for multivariate polynomial interpolation

Author:

Chen Jianxin1,Childs Andrew M.231ORCID,Hung Shih-Han21ORCID

Affiliation:

1. Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD 20742, USA

2. Department of Computer Science, University of Maryland, College Park, MD 20742, USA

3. Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742, USA

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 F q , R and C . We show that k C and 2 k C queries suffice to achieve probability 1 for C and R , respectively, where k C = ( 1 / ( n + 1 ) ) ( n + d d ) except for d =2 and four other special cases. For F q , we show that ⌈( d /( n + d ))( n + d d ) ⌉ 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 speed-up by a factor of n +1, ( n +1)/2 and ( n + d )/ d for C , R and 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 F q , we conjecture that 2 k C queries also suffice to achieve probability approaching 1 for large field order q , although we leave this as an open problem.

Funder

U.S. Department of Defense

National Science Foundation

Canadian Institute for Advanced Research

Publisher

The Royal Society

Subject

General Physics and Astronomy,General Engineering,General Mathematics

Reference27 articles.

1. ChildsAM van DamW HungS-H ShparlinskiIE. 2016 Optimal quantum algorithm for polynomial interpolation. In 43rd International Colloquium on Automata Languages and Programming (ICALP 2016) . Leibniz International Proceedings in Informatics vol. 55 pp. 16:1–16:13 Dagstuhl Germany: Schloss Dagstuhl-Liebniz-Zentrum fuer Informatik. Available at arXiv:1509.09271.

2. Quantum interpolation of polynomials;Kane DM;Quantum Inf. Comput.,2011

3. On the uselessness of quantum queries

4. The quantum query complexity of learning multilinear polynomials

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