Optimal quantum algorithm for polynomial interpolation
Abstract
We consider the number of quantum queries required to determine the coefficients of a degree-d polynomial over GF(q). A lower bound shown independently by Kane and Kutin and by Meyer and Pommersheim shows that d/2+1/2 quantum queries are needed to solve this problem with bounded error, whereas an algorithm of Boneh and Zhandry shows that d quantum queries are sufficient. We show that the lower bound is achievable: d/2+1/2 quantum queries suffice to determine the polynomial with bounded error. Furthermore, we show that d/2+1 queries suffice to achieve probability approaching 1 for large q. These upper bounds improve results of Boneh and Zhandry on the insecurity of cryptographic protocols against quantum attacks. We also show that our algorithm's success probability as a function of the number of queries is precisely optimal. Furthermore, the algorithm can be implemented with gate complexity poly(log q) with negligible decrease in the success probability. We end with a conjecture about the quantum query complexity of multivariate polynomial interpolation.
Keywords
Cite
@article{arxiv.1509.09271,
title = {Optimal quantum algorithm for polynomial interpolation},
author = {Andrew M. Childs and Wim van Dam and Shih-Han Hung and Igor E. Shparlinski},
journal= {arXiv preprint arXiv:1509.09271},
year = {2016}
}
Comments
17 pages, minor improvements, added conjecture about multivariate interpolation