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&⋕39;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.
Publication Details
- Authors
- Publication Type
- Journal Article
- Year of Publication
- 2016
- Journal
- 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)
- Volume
- 55
- Date Published
- 03/2016
- Pagination
- 16:1–16:13
- ISSN
- 1868-8969